| L(s) = 1 | − 2·2-s − 4-s − 7-s + 8·8-s − 5·9-s − 4·11-s + 2·14-s − 7·16-s + 10·18-s + 8·22-s + 4·23-s − 25-s + 28-s − 12·29-s − 14·32-s + 5·36-s − 4·37-s + 8·43-s + 4·44-s − 8·46-s − 6·49-s + 2·50-s + 16·53-s − 8·56-s + 24·58-s + 5·63-s + 35·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 0.377·7-s + 2.82·8-s − 5/3·9-s − 1.20·11-s + 0.534·14-s − 7/4·16-s + 2.35·18-s + 1.70·22-s + 0.834·23-s − 1/5·25-s + 0.188·28-s − 2.22·29-s − 2.47·32-s + 5/6·36-s − 0.657·37-s + 1.21·43-s + 0.603·44-s − 1.17·46-s − 6/7·49-s + 0.282·50-s + 2.19·53-s − 1.06·56-s + 3.15·58-s + 0.629·63-s + 35/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.487809841781377994708404283250, −8.282431792555002607496667489786, −7.998948016327729487855897110735, −7.29956833629202039981876932842, −7.12147698344619796195085066396, −6.19328726666763048117092984337, −5.46751543572783023109461357627, −5.29917071437467895295333587611, −4.90809995513951539087505569150, −3.76592143771251579951082054976, −3.74704903414376473785152527415, −2.64832298571961526499693553463, −2.04565274383482105934865523448, −0.77072642791312963808908143508, 0,
0.77072642791312963808908143508, 2.04565274383482105934865523448, 2.64832298571961526499693553463, 3.74704903414376473785152527415, 3.76592143771251579951082054976, 4.90809995513951539087505569150, 5.29917071437467895295333587611, 5.46751543572783023109461357627, 6.19328726666763048117092984337, 7.12147698344619796195085066396, 7.29956833629202039981876932842, 7.998948016327729487855897110735, 8.282431792555002607496667489786, 8.487809841781377994708404283250
