| L(s) = 1 | − 2.03e3·7-s + 4.23e3·9-s + 2.94e4·17-s − 1.37e5·23-s + 1.12e5·25-s + 4.55e5·31-s − 2.16e4·41-s + 9.45e5·47-s + 1.44e6·49-s − 8.59e6·63-s + 2.82e6·71-s − 1.96e6·73-s − 7.13e6·79-s + 1.31e7·81-s + 2.39e7·89-s + 1.73e7·97-s − 7.49e6·103-s + 3.31e7·113-s − 5.97e7·119-s + 3.77e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.24e8·153-s + ⋯ |
| L(s) = 1 | − 2.23·7-s + 1.93·9-s + 1.45·17-s − 2.35·23-s + 1.43·25-s + 2.74·31-s − 0.0491·41-s + 1.32·47-s + 1.76·49-s − 4.33·63-s + 0.938·71-s − 0.589·73-s − 1.62·79-s + 2.74·81-s + 3.59·89-s + 1.93·97-s − 0.675·103-s + 2.16·113-s − 3.25·119-s + 1.93·121-s + 2.80·153-s + 5.27·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.671574270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.671574270\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 470 p^{2} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4486 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 1016 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 37781878 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 123587110 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14706 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 192539878 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 68712 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 23979147718 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 227552 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 164095157590 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10842 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 145794182710 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 472656 T + p^{7} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 117332495350 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1995782265962 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5600395071238 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12105546202630 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 1414728 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 980282 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3566800 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22090398335590 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11951190 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8682146 T + p^{7} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69143124889607711280144845974, −10.19649227265290576775735671873, −9.956794140406811873933314442279, −9.955858273919529335587875429547, −9.260800378263687040734122727387, −8.706389826952533892518894554517, −7.82562463532229256615577362083, −7.71067356140052010232383583258, −6.81518232650517332970813831163, −6.66678791921661030951050965831, −6.11711903786349302302114905355, −5.67504036366272244353794261810, −4.59023998422853910749998102577, −4.39146098380904090063654521110, −3.47977568626439206832395231988, −3.32372884683892757623712627467, −2.49928753085013537770277691351, −1.74618937612489172208654345767, −0.77322912813598430577122926748, −0.65149990697197574997933706823,
0.65149990697197574997933706823, 0.77322912813598430577122926748, 1.74618937612489172208654345767, 2.49928753085013537770277691351, 3.32372884683892757623712627467, 3.47977568626439206832395231988, 4.39146098380904090063654521110, 4.59023998422853910749998102577, 5.67504036366272244353794261810, 6.11711903786349302302114905355, 6.66678791921661030951050965831, 6.81518232650517332970813831163, 7.71067356140052010232383583258, 7.82562463532229256615577362083, 8.706389826952533892518894554517, 9.260800378263687040734122727387, 9.955858273919529335587875429547, 9.956794140406811873933314442279, 10.19649227265290576775735671873, 10.69143124889607711280144845974
