| L(s) = 1 | − 4·2-s + 12·4-s − 32·8-s − 4·11-s + 80·16-s + 16·22-s − 60·23-s − 18·25-s − 424·29-s − 192·32-s + 492·37-s − 568·43-s − 48·44-s + 240·46-s + 72·50-s − 1.09e3·53-s + 1.69e3·58-s + 448·64-s + 1.30e3·67-s − 1.54e3·71-s − 1.96e3·74-s + 944·79-s + 2.27e3·86-s + 128·88-s − 720·92-s − 216·100-s + 4.38e3·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.109·11-s + 5/4·16-s + 0.155·22-s − 0.543·23-s − 0.143·25-s − 2.71·29-s − 1.06·32-s + 2.18·37-s − 2.01·43-s − 0.164·44-s + 0.769·46-s + 0.203·50-s − 2.84·53-s + 3.83·58-s + 7/8·64-s + 2.37·67-s − 2.57·71-s − 3.09·74-s + 1.34·79-s + 2.84·86-s + 0.155·88-s − 0.815·92-s − 0.215·100-s + 4.01·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 18 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3466 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7738 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 9482 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 212 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 14110 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 35530 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 284 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 203934 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 548 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 38394 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 185770 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 652 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 770 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 172238 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 472 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1110166 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 897450 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1732546 T^{2} + p^{6} T^{4} \) |
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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
−9.452793544717196222174419048355, −9.276940348827369100559630365676, −8.730775018060954274199349119871, −8.291769713180933081318692718202, −7.76843512357802153775511473467, −7.65300756128186136920200375570, −7.22637541130670793683163750927, −6.53122590028163273720299836186, −6.15802044935025213151043510973, −5.92634072868277291425859215476, −5.08117333641296461362158615117, −4.82761472498485859550376710411, −3.72517154864632884131968303331, −3.69709598333331699933528627807, −2.78858336596011859982756794418, −2.25978069439975492284996951056, −1.66004194316077754044389828864, −1.15077560373834186920809755673, 0, 0,
1.15077560373834186920809755673, 1.66004194316077754044389828864, 2.25978069439975492284996951056, 2.78858336596011859982756794418, 3.69709598333331699933528627807, 3.72517154864632884131968303331, 4.82761472498485859550376710411, 5.08117333641296461362158615117, 5.92634072868277291425859215476, 6.15802044935025213151043510973, 6.53122590028163273720299836186, 7.22637541130670793683163750927, 7.65300756128186136920200375570, 7.76843512357802153775511473467, 8.291769713180933081318692718202, 8.730775018060954274199349119871, 9.276940348827369100559630365676, 9.452793544717196222174419048355
