| L(s) = 1 | + 4·2-s + 6·4-s + 8·8-s − 9-s + 2·11-s + 9·16-s − 4·18-s + 8·22-s + 24·23-s + 7·25-s + 2·29-s − 24·32-s − 6·36-s + 40·37-s + 14·43-s + 12·44-s + 96·46-s + 28·50-s − 24·53-s + 8·58-s − 86·64-s − 4·67-s − 16·71-s − 8·72-s + 160·74-s + 12·79-s + 15·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 3·4-s + 2.82·8-s − 1/3·9-s + 0.603·11-s + 9/4·16-s − 0.942·18-s + 1.70·22-s + 5.00·23-s + 7/5·25-s + 0.371·29-s − 4.24·32-s − 36-s + 6.57·37-s + 2.13·43-s + 1.80·44-s + 14.1·46-s + 3.95·50-s − 3.29·53-s + 1.05·58-s − 10.7·64-s − 0.488·67-s − 1.89·71-s − 0.942·72-s + 18.5·74-s + 1.35·79-s + 5/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(42.67253978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(42.67253978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + p T^{4} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 3 | \( 1 + T^{2} - 14 T^{4} - p T^{6} + 5 p^{3} T^{8} - p^{3} T^{10} - 14 p^{4} T^{12} + p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 7 T^{2} - 2 p T^{4} - 63 T^{6} + 1631 T^{8} - 63 p^{2} T^{10} - 2 p^{5} T^{12} - 7 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - T + 8 T^{2} + 29 T^{3} - 83 T^{4} + 29 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 16 T^{2} + 5 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 63 T^{2} + 2258 T^{4} - 62307 T^{6} + 1364391 T^{8} - 62307 p^{2} T^{10} + 2258 p^{4} T^{12} - 63 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - T - 54 T^{2} + 3 T^{3} + 2155 T^{4} + 3 p T^{5} - 54 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 72 T^{2} + 2603 T^{4} - 47448 T^{6} + 756216 T^{8} - 47448 p^{2} T^{10} + 2603 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 10 T + 86 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 8 T^{2} - 2846 T^{4} + 3616 T^{6} + 5534755 T^{8} + 3616 p^{2} T^{10} - 2846 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 7 T + 32 T^{2} + 483 T^{3} - 3667 T^{4} + 483 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 32 T^{2} + 2083 T^{4} + 175264 T^{6} - 6020216 T^{8} + 175264 p^{2} T^{10} + 2083 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T + 15 T^{2} + 276 T^{3} + 6200 T^{4} + 276 p T^{5} + 15 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 184 T^{2} + 15101 T^{4} + 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 - 192 T^{2} + 20258 T^{4} - 1759488 T^{6} + 124742451 T^{8} - 1759488 p^{2} T^{10} + 20258 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 240 T^{2} + 32594 T^{4} - 3443520 T^{6} + 289951395 T^{8} - 3443520 p^{2} T^{10} + 32594 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 6 T - 118 T^{2} + 24 T^{3} + 14631 T^{4} + 24 p T^{5} - 118 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 280 T^{2} + 33053 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 239 T^{2} + 29859 T^{4} + 239 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 141 T^{2} + 74 T^{4} - 139449 T^{6} + 116727639 T^{8} - 139449 p^{2} T^{10} + 74 p^{4} T^{12} - 141 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.70793841751802662985115601847, −4.54513736885232776112742573787, −4.41157052565804370679655353957, −4.21614897193416035466260421634, −4.14096778828151628742803287955, −4.03969752380130787483781398091, −3.91763508375005680301039835934, −3.85162376673091727638484744433, −3.48289957851124662758631142592, −3.34516918117579994451855229286, −3.21103067885206806309992634717, −3.09662355200306867031010458569, −3.00539110293087807965607387180, −2.76141597478575281667236073436, −2.67138460481525229414154562990, −2.55098128285748599109124461407, −2.37293705125911309265108419578, −2.10755571331038405175878353261, −1.89143819262262477746876585991, −1.67215533806631131637059348646, −1.56721718190320522374654899581, −1.02926245911253534949341727189, −0.937295552965513033568204379849, −0.885971315639724388429662260458, −0.56274131094727279762342466635,
0.56274131094727279762342466635, 0.885971315639724388429662260458, 0.937295552965513033568204379849, 1.02926245911253534949341727189, 1.56721718190320522374654899581, 1.67215533806631131637059348646, 1.89143819262262477746876585991, 2.10755571331038405175878353261, 2.37293705125911309265108419578, 2.55098128285748599109124461407, 2.67138460481525229414154562990, 2.76141597478575281667236073436, 3.00539110293087807965607387180, 3.09662355200306867031010458569, 3.21103067885206806309992634717, 3.34516918117579994451855229286, 3.48289957851124662758631142592, 3.85162376673091727638484744433, 3.91763508375005680301039835934, 4.03969752380130787483781398091, 4.14096778828151628742803287955, 4.21614897193416035466260421634, 4.41157052565804370679655353957, 4.54513736885232776112742573787, 4.70793841751802662985115601847
Plot not available for L-functions of degree greater than 10.
