| 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
| \(L(4)\) |
\(\approx\) |
\(2.400245560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400245560\) |
| \(L(\frac{9}{2})\) |
|
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
−10.97734890127892479495126572898, −10.66906969307949070777471613965, −10.11831912229091367840177474291, −9.365883977455520315836849266810, −9.076568408744682903115659674472, −8.656889447820584858065578686080, −8.388551218446734898895550144371, −7.41712254529445219031962533083, −6.83916982901611453655435057768, −6.77294091715555615425052614033, −5.99199671231005306230123201707, −5.52457963273437033508747688568, −4.77293247286325986846104780945, −4.53184309673845843615059793608, −3.40843528008567585308453970880, −2.93857860247985236165877282021, −2.80368740528986447017807180074, −1.79989325446240354791157743987, −0.790579660537015334073372346359, −0.49445813957090068385499906448,
0.49445813957090068385499906448, 0.790579660537015334073372346359, 1.79989325446240354791157743987, 2.80368740528986447017807180074, 2.93857860247985236165877282021, 3.40843528008567585308453970880, 4.53184309673845843615059793608, 4.77293247286325986846104780945, 5.52457963273437033508747688568, 5.99199671231005306230123201707, 6.77294091715555615425052614033, 6.83916982901611453655435057768, 7.41712254529445219031962533083, 8.388551218446734898895550144371, 8.656889447820584858065578686080, 9.076568408744682903115659674472, 9.365883977455520315836849266810, 10.11831912229091367840177474291, 10.66906969307949070777471613965, 10.97734890127892479495126572898
