| L(s) = 1 | + 16·4-s + 52·7-s − 572·13-s + 64·16-s − 248·19-s − 304·25-s + 832·28-s − 3.58e3·31-s − 1.88e4·37-s − 3.02e3·43-s + 1.49e3·49-s − 9.15e3·52-s + 4.14e3·61-s − 1.02e3·64-s + 1.20e4·67-s − 3.58e3·73-s − 3.96e3·76-s + 1.50e4·79-s − 2.97e4·91-s − 4.63e4·97-s − 4.86e3·100-s + 3.35e4·103-s + 8.90e4·109-s + 3.32e3·112-s − 4.42e4·121-s − 5.73e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s + 1.06·7-s − 3.38·13-s + 1/4·16-s − 0.686·19-s − 0.486·25-s + 1.06·28-s − 3.72·31-s − 13.7·37-s − 1.63·43-s + 0.621·49-s − 3.38·52-s + 1.11·61-s − 1/4·64-s + 2.69·67-s − 0.672·73-s − 0.686·76-s + 2.40·79-s − 3.59·91-s − 4.92·97-s − 0.486·100-s + 3.16·103-s + 7.49·109-s + 0.265·112-s − 3.01·121-s − 3.72·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1423539991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1423539991\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{3} T^{2} + p^{6} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 304 T^{2} + 314737 T^{4} - 305085584 T^{6} - 139261263584 T^{8} - 305085584 p^{8} T^{10} + 314737 p^{16} T^{12} + 304 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 - 26 T + 268 T^{2} + 114244 T^{3} - 7221413 T^{4} + 114244 p^{4} T^{5} + 268 p^{8} T^{6} - 26 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 + 44200 T^{2} + 1068005038 T^{4} + 20195740240000 T^{6} + 319619471776329283 T^{8} + 20195740240000 p^{8} T^{10} + 1068005038 p^{16} T^{12} + 44200 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 22 p T + 9517 T^{2} + 333454 p T^{3} + 2072347564 T^{4} + 333454 p^{5} T^{5} + 9517 p^{8} T^{6} + 22 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( ( 1 - 267736 T^{2} + 30823871559 T^{4} - 267736 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 19 | \( ( 1 + 62 T + 211680 T^{2} + 62 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 23 | \( 1 - 33536 T^{2} - 77046473090 T^{4} + 2630927174926336 T^{6} - \)\(10\!\cdots\!41\)\( T^{8} + 2630927174926336 p^{8} T^{10} - 77046473090 p^{16} T^{12} - 33536 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 + 343288 T^{2} - 592173740039 T^{4} - 99715701245339432 T^{6} + \)\(19\!\cdots\!84\)\( T^{8} - 99715701245339432 p^{8} T^{10} - 592173740039 p^{16} T^{12} + 343288 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 + 1792 T + 640138 T^{2} + 1297558528 T^{3} + 2792128129891 T^{4} + 1297558528 p^{4} T^{5} + 640138 p^{8} T^{6} + 1792 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4700 T + 9270795 T^{2} + 4700 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 41 | \( 1 + 8941048 T^{2} + 44849615538094 T^{4} + \)\(17\!\cdots\!64\)\( T^{6} + \)\(52\!\cdots\!75\)\( T^{8} + \)\(17\!\cdots\!64\)\( p^{8} T^{10} + 44849615538094 p^{16} T^{12} + 8941048 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 + 1510 T - 4281644 T^{2} - 416545580 T^{3} + 23573075883835 T^{4} - 416545580 p^{4} T^{5} - 4281644 p^{8} T^{6} + 1510 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 + 7387156 T^{2} + 18121663956442 T^{4} - 82545289029903849968 T^{6} - \)\(53\!\cdots\!81\)\( T^{8} - 82545289029903849968 p^{8} T^{10} + 18121663956442 p^{16} T^{12} + 7387156 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( ( 1 - 28489720 T^{2} + 325176478293714 T^{4} - 28489720 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 59 | \( 1 + 44516212 T^{2} + 1194314285673178 T^{4} + \)\(21\!\cdots\!88\)\( T^{6} + \)\(30\!\cdots\!91\)\( T^{8} + \)\(21\!\cdots\!88\)\( p^{8} T^{10} + 1194314285673178 p^{16} T^{12} + 44516212 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 2072 T - 21145367 T^{2} + 4668487432 T^{3} + 395969820934576 T^{4} + 4668487432 p^{4} T^{5} - 21145367 p^{8} T^{6} - 2072 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 - 6038 T + 21578188 T^{2} + 153503989468 T^{3} - 919333783837829 T^{4} + 153503989468 p^{4} T^{5} + 21578188 p^{8} T^{6} - 6038 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( ( 1 - 26476960 T^{2} + 845007854749314 T^{4} - 26476960 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 73 | \( ( 1 + 896 T - 162489 T^{2} + 896 p^{4} T^{3} + p^{8} T^{4} )^{4} \) |
| 79 | \( ( 1 - 7502 T - 35614316 T^{2} - 104984173316 T^{3} + 4735346523875515 T^{4} - 104984173316 p^{4} T^{5} - 35614316 p^{8} T^{6} - 7502 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 + 185096692 T^{2} + 21192092541154858 T^{4} + \)\(15\!\cdots\!08\)\( T^{6} + \)\(88\!\cdots\!31\)\( T^{8} + \)\(15\!\cdots\!08\)\( p^{8} T^{10} + 21192092541154858 p^{16} T^{12} + 185096692 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( ( 1 - 116789296 T^{2} + 8841733583204319 T^{4} - 116789296 p^{8} T^{6} + p^{16} T^{8} )^{2} \) |
| 97 | \( ( 1 + 23152 T + 227547466 T^{2} + 3042382927552 T^{3} + 39861818336661859 T^{4} + 3042382927552 p^{4} T^{5} + 227547466 p^{8} T^{6} + 23152 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.17391955478848289890213021080, −5.15669432275966114804094319915, −5.11260245857849206341162471118, −4.79982622171108095295116690168, −4.63878414352447064977494468098, −4.04934378741390053639303379918, −4.02687955801441480136905574007, −3.85072386453772780672842706826, −3.65326343234008419766652731186, −3.58845529223399943831746453426, −3.41130791316374760248805599604, −3.28135416269961450335083021817, −3.04033118768955115601207867377, −2.68161791228087572915073556520, −2.46671011225977705969861453771, −2.17048088875929241793692025480, −2.01184541473869320880390228628, −1.84751062786432203001577928666, −1.70035321516695241219496604682, −1.70018912987514863128351857132, −1.69094880290039602065955543483, −1.06693972246287747964692662658, −0.35420835029935112297326025529, −0.23352924314046519199533659772, −0.088888185955104982266063468310,
0.088888185955104982266063468310, 0.23352924314046519199533659772, 0.35420835029935112297326025529, 1.06693972246287747964692662658, 1.69094880290039602065955543483, 1.70018912987514863128351857132, 1.70035321516695241219496604682, 1.84751062786432203001577928666, 2.01184541473869320880390228628, 2.17048088875929241793692025480, 2.46671011225977705969861453771, 2.68161791228087572915073556520, 3.04033118768955115601207867377, 3.28135416269961450335083021817, 3.41130791316374760248805599604, 3.58845529223399943831746453426, 3.65326343234008419766652731186, 3.85072386453772780672842706826, 4.02687955801441480136905574007, 4.04934378741390053639303379918, 4.63878414352447064977494468098, 4.79982622171108095295116690168, 5.11260245857849206341162471118, 5.15669432275966114804094319915, 5.17391955478848289890213021080
Plot not available for L-functions of degree greater than 10.
