| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 6·5-s + 4·6-s + 3·9-s + 12·10-s − 4·12-s + 12·15-s − 4·16-s − 2·17-s − 6·18-s − 19-s − 12·20-s + 17·25-s − 4·27-s − 24·30-s − 12·31-s + 8·32-s + 4·34-s + 6·36-s + 2·38-s − 18·45-s + 8·48-s + 11·49-s − 34·50-s + 4·51-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 2.68·5-s + 1.63·6-s + 9-s + 3.79·10-s − 1.15·12-s + 3.09·15-s − 16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s − 2.68·20-s + 17/5·25-s − 0.769·27-s − 4.38·30-s − 2.15·31-s + 1.41·32-s + 0.685·34-s + 36-s + 0.324·38-s − 2.68·45-s + 1.15·48-s + 11/7·49-s − 4.80·50-s + 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.48919464134144900735099782123, −7.47848778675296589128891353641, −7.27079151643991129277821474463, −6.41173993136661837405396202601, −6.25499216179811745555302525608, −5.36600607918508759890630649285, −4.89390739159963849894021052339, −4.42808617434113254239775941766, −3.81591641511052084615077615666, −3.79824334094358549621418265757, −2.80250883164613859029957691965, −1.86642316096709822967327164489, −1.02602590121900935054247645611, 0, 0,
1.02602590121900935054247645611, 1.86642316096709822967327164489, 2.80250883164613859029957691965, 3.79824334094358549621418265757, 3.81591641511052084615077615666, 4.42808617434113254239775941766, 4.89390739159963849894021052339, 5.36600607918508759890630649285, 6.25499216179811745555302525608, 6.41173993136661837405396202601, 7.27079151643991129277821474463, 7.47848778675296589128891353641, 7.48919464134144900735099782123
