| L(s) = 1 | − 2·2-s − 4-s − 2·7-s + 8·8-s − 6·9-s − 12·11-s + 4·14-s − 7·16-s + 12·18-s + 24·22-s + 4·23-s + 25-s + 2·28-s − 2·29-s − 14·32-s + 6·36-s + 20·37-s + 16·43-s + 12·44-s − 8·46-s − 3·49-s − 2·50-s − 12·53-s − 16·56-s + 4·58-s + 12·63-s + 35·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 0.755·7-s + 2.82·8-s − 2·9-s − 3.61·11-s + 1.06·14-s − 7/4·16-s + 2.82·18-s + 5.11·22-s + 0.834·23-s + 1/5·25-s + 0.377·28-s − 0.371·29-s − 2.47·32-s + 36-s + 3.28·37-s + 2.43·43-s + 1.80·44-s − 1.17·46-s − 3/7·49-s − 0.282·50-s − 1.64·53-s − 2.13·56-s + 0.525·58-s + 1.51·63-s + 35/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1030225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1030225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.66674201908184981995831191267, −7.64947192507391358342580089136, −7.28900292042912777665088054536, −6.01956800466047829467852819351, −6.01238272522846614289743699282, −5.39388342342135590297843182122, −5.00289399095436617882638449140, −4.64971827708119930945851978376, −3.99163533960682761754349881016, −2.95203182599403093553876063934, −2.83663453400435515061927081115, −2.37114931223056009945275538300, −0.993298978531287449348578624970, 0, 0,
0.993298978531287449348578624970, 2.37114931223056009945275538300, 2.83663453400435515061927081115, 2.95203182599403093553876063934, 3.99163533960682761754349881016, 4.64971827708119930945851978376, 5.00289399095436617882638449140, 5.39388342342135590297843182122, 6.01238272522846614289743699282, 6.01956800466047829467852819351, 7.28900292042912777665088054536, 7.64947192507391358342580089136, 7.66674201908184981995831191267
