| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 2·11-s − 6·13-s + 14-s + 16-s + 4·17-s − 6·19-s − 2·22-s − 8·23-s − 6·26-s + 28-s − 6·29-s − 2·31-s + 32-s + 4·34-s − 4·37-s − 6·38-s − 2·41-s − 4·43-s − 2·44-s − 8·46-s + 8·47-s + 49-s − 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 1.37·19-s − 0.426·22-s − 1.66·23-s − 1.17·26-s + 0.188·28-s − 1.11·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.973·38-s − 0.312·41-s − 0.609·43-s − 0.301·44-s − 1.17·46-s + 1.16·47-s + 1/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.036835570616111132687795568095, −7.57781723598090862974782182719, −6.81080374028761293998094873350, −5.78793574564528183106350649884, −5.27167215176390503105650672282, −4.43244936590476019100850584085, −3.67274830953119673001999303428, −2.51052952009881589815450760658, −1.88342392731401272278008367686, 0,
1.88342392731401272278008367686, 2.51052952009881589815450760658, 3.67274830953119673001999303428, 4.43244936590476019100850584085, 5.27167215176390503105650672282, 5.78793574564528183106350649884, 6.81080374028761293998094873350, 7.57781723598090862974782182719, 8.036835570616111132687795568095
