| L(s) = 1 | − 2-s + 4-s + 5-s − 3·7-s − 8-s − 10-s + 4·11-s + 13-s + 3·14-s + 16-s + 3·17-s − 19-s + 20-s − 4·22-s − 2·23-s − 4·25-s − 26-s − 3·28-s − 4·29-s − 8·31-s − 32-s − 3·34-s − 3·35-s + 37-s + 38-s − 40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.13·7-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.277·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s − 0.852·22-s − 0.417·23-s − 4/5·25-s − 0.196·26-s − 0.566·28-s − 0.742·29-s − 1.43·31-s − 0.176·32-s − 0.514·34-s − 0.507·35-s + 0.164·37-s + 0.162·38-s − 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.033392300041384382264664293095, −7.23458889747962575651296066005, −6.53572069080007095288393890464, −6.04195854869870819744741686032, −5.26893580918868684044099399400, −3.81497238059810570995938897834, −3.47546022166629546985070789532, −2.21769652204214886580824677712, −1.35814928467585276435922401808, 0,
1.35814928467585276435922401808, 2.21769652204214886580824677712, 3.47546022166629546985070789532, 3.81497238059810570995938897834, 5.26893580918868684044099399400, 6.04195854869870819744741686032, 6.53572069080007095288393890464, 7.23458889747962575651296066005, 8.033392300041384382264664293095
