| L(s) = 1 | + 912·7-s − 2.68e3·9-s − 2.23e4·17-s − 1.63e5·23-s + 1.49e5·25-s − 8.09e4·31-s − 2.82e5·41-s − 1.36e6·47-s − 1.02e6·49-s − 2.44e6·63-s + 5.09e6·71-s + 3.36e6·73-s + 8.07e6·79-s + 2.41e6·81-s + 1.29e7·89-s − 1.21e7·97-s − 8.20e6·103-s + 1.86e7·113-s − 2.03e7·119-s + 3.26e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 5.98e7·153-s + ⋯ |
| L(s) = 1 | + 1.00·7-s − 1.22·9-s − 1.10·17-s − 2.80·23-s + 1.91·25-s − 0.488·31-s − 0.640·41-s − 1.91·47-s − 1.24·49-s − 1.23·63-s + 1.68·71-s + 1.01·73-s + 1.84·79-s + 0.503·81-s + 1.94·89-s − 1.34·97-s − 0.739·103-s + 1.21·113-s − 1.10·119-s + 1.67·121-s + 1.35·153-s − 2.81·161-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
| \(L(4)\) |
\(\approx\) |
\(0.4490013966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4490013966\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 + 298 p^{2} T^{2} + p^{14} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 149526 T^{2} + p^{14} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 456 T + p^{7} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 32603766 T^{2} + p^{14} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 9331750 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 11150 T + p^{7} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 1770736102 T^{2} + p^{14} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 81704 T + p^{7} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 24540111814 T^{2} + p^{14} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 40480 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 13932162902 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 141402 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 66946399030 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 682032 T + p^{7} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 937974602250 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4044092872854 T^{2} + p^{14} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2722187643142 T^{2} + p^{14} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3325050217222 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2548232 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1680326 T + p^{7} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4038064 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 25265648115558 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6473046 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6065758 T + p^{7} T^{2} )^{2} \) |
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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
−11.14925570086343818736714994976, −10.72532125753540279067780307607, −10.04218243679370980611432845593, −9.577656471168676227115989819965, −9.004117622771594297649082735016, −8.453648364449749936029808142988, −8.105087292105323844963876075342, −7.921895020352833980086120407616, −6.98445456768812272635542563173, −6.40310702394613329366114798368, −6.17195982811732043725960572218, −5.24091705598414179955281739247, −5.01206895375848133957518027066, −4.40262392439682416425100183962, −3.64416350855205688470013100548, −3.12577401834294823773825107298, −2.12233151472190567563808185408, −2.05363110490017659800337975830, −1.09218341792945546554962518828, −0.16054144682128735812978866171,
0.16054144682128735812978866171, 1.09218341792945546554962518828, 2.05363110490017659800337975830, 2.12233151472190567563808185408, 3.12577401834294823773825107298, 3.64416350855205688470013100548, 4.40262392439682416425100183962, 5.01206895375848133957518027066, 5.24091705598414179955281739247, 6.17195982811732043725960572218, 6.40310702394613329366114798368, 6.98445456768812272635542563173, 7.921895020352833980086120407616, 8.105087292105323844963876075342, 8.453648364449749936029808142988, 9.004117622771594297649082735016, 9.577656471168676227115989819965, 10.04218243679370980611432845593, 10.72532125753540279067780307607, 11.14925570086343818736714994976
