| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s − 11-s + 4·13-s − 3·14-s + 16-s − 3·17-s − 5·19-s + 22-s − 4·23-s − 4·26-s + 3·28-s − 5·29-s + 7·31-s − 32-s + 3·34-s − 7·37-s + 5·38-s + 8·41-s − 6·43-s − 44-s + 4·46-s − 8·47-s + 2·49-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s − 0.301·11-s + 1.10·13-s − 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.14·19-s + 0.213·22-s − 0.834·23-s − 0.784·26-s + 0.566·28-s − 0.928·29-s + 1.25·31-s − 0.176·32-s + 0.514·34-s − 1.15·37-s + 0.811·38-s + 1.24·41-s − 0.914·43-s − 0.150·44-s + 0.589·46-s − 1.16·47-s + 2/7·49-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.120191452422177114432965987488, −7.38155858128562355916640707717, −6.40550294377758213402642700076, −5.97134943250727568183682660114, −4.86461296537077672836657233082, −4.24258949392965466519512212389, −3.19555527256937768583664502196, −2.04894811678546610668362989566, −1.47243511430493457189804457763, 0,
1.47243511430493457189804457763, 2.04894811678546610668362989566, 3.19555527256937768583664502196, 4.24258949392965466519512212389, 4.86461296537077672836657233082, 5.97134943250727568183682660114, 6.40550294377758213402642700076, 7.38155858128562355916640707717, 8.120191452422177114432965987488
