| L(s) = 1 | + 1.22e3·17-s + 1.32e3·25-s − 1.18e4·41-s + 2.87e3·49-s − 2.35e4·73-s − 3.50e4·89-s + 2.36e4·97-s − 6.14e4·113-s − 5.81e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.13e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4.23·17-s + 2.11·25-s − 7.06·41-s + 1.19·49-s − 4.42·73-s − 4.42·89-s + 2.51·97-s − 4.80·113-s − 3.97·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.96·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.553019086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553019086\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 662 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 1438 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 29090 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 56690 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 p T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 102670 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 340658 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 732586 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1834942 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2668322 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2970 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 1509070 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 9602546 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 14513462 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 17045810 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8140126 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 17250290 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7421618 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 5894 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 5887966 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94916450 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8766 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5918 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96853020118629211389195352374, −6.89926591036386455210925821746, −6.74935382347212281887900576114, −6.52685681212796169166746256605, −6.05334778555968845348108606937, −5.75912214044975777687463040411, −5.71707568254131179888104616009, −5.19108827014840352275216159500, −5.06387795348541500967621120396, −5.05039813869849369966718501602, −5.02888363041396077347805217448, −4.19854130125694695183558640409, −3.93296366845340768884598355175, −3.87807082957901220294997195080, −3.43941429242788714875623886050, −3.07772995700007501562644237282, −3.05382735755918681139550269290, −2.81874914104307685120001305152, −2.50603213485441464834279846211, −1.64889686292710059698867690061, −1.43059986297073727279643692032, −1.34453268586427533506611922395, −1.30226530012618004123728893309, −0.53589296286443654262950760966, −0.15988433332669184288079933666,
0.15988433332669184288079933666, 0.53589296286443654262950760966, 1.30226530012618004123728893309, 1.34453268586427533506611922395, 1.43059986297073727279643692032, 1.64889686292710059698867690061, 2.50603213485441464834279846211, 2.81874914104307685120001305152, 3.05382735755918681139550269290, 3.07772995700007501562644237282, 3.43941429242788714875623886050, 3.87807082957901220294997195080, 3.93296366845340768884598355175, 4.19854130125694695183558640409, 5.02888363041396077347805217448, 5.05039813869849369966718501602, 5.06387795348541500967621120396, 5.19108827014840352275216159500, 5.71707568254131179888104616009, 5.75912214044975777687463040411, 6.05334778555968845348108606937, 6.52685681212796169166746256605, 6.74935382347212281887900576114, 6.89926591036386455210925821746, 6.96853020118629211389195352374
