| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·9-s − 4·11-s + 5·16-s − 8·18-s − 8·22-s − 10·25-s − 12·29-s + 6·32-s − 12·36-s − 4·37-s − 12·43-s − 12·44-s − 20·50-s + 2·53-s − 24·58-s + 7·64-s + 8·67-s + 16·71-s − 16·72-s − 8·74-s − 24·79-s + 7·81-s − 24·86-s − 16·88-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s − 1.20·11-s + 5/4·16-s − 1.88·18-s − 1.70·22-s − 2·25-s − 2.22·29-s + 1.06·32-s − 2·36-s − 0.657·37-s − 1.82·43-s − 1.80·44-s − 2.82·50-s + 0.274·53-s − 3.15·58-s + 7/8·64-s + 0.977·67-s + 1.89·71-s − 1.88·72-s − 0.929·74-s − 2.70·79-s + 7/9·81-s − 2.58·86-s − 1.70·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26977636 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26977636 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 53 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−7.907391988025840112195600170615, −7.73484004825153356705822525206, −7.10031646213773678597591475408, −7.03881478550339280851073322932, −6.42268292566425518859325936842, −6.07703208479603311896104854643, −5.63487520072029528124312462262, −5.53806868170035424956252457516, −5.16444369534845033467509812323, −4.93720037988371086196588641170, −4.19528812608915231434809424783, −3.94663319521752276784553730279, −3.37625594198578000932538242992, −3.37266344297077226701821464763, −2.54472466158891593617956756647, −2.41469691043021875125231859681, −1.89710798323623730877736978384, −1.37233738347896103159440792186, 0, 0,
1.37233738347896103159440792186, 1.89710798323623730877736978384, 2.41469691043021875125231859681, 2.54472466158891593617956756647, 3.37266344297077226701821464763, 3.37625594198578000932538242992, 3.94663319521752276784553730279, 4.19528812608915231434809424783, 4.93720037988371086196588641170, 5.16444369534845033467509812323, 5.53806868170035424956252457516, 5.63487520072029528124312462262, 6.07703208479603311896104854643, 6.42268292566425518859325936842, 7.03881478550339280851073322932, 7.10031646213773678597591475408, 7.73484004825153356705822525206, 7.907391988025840112195600170615
