| L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s − 2·10-s − 4·11-s + 12-s + 13-s + 4·14-s + 2·15-s − 16-s − 2·17-s + 18-s + 2·20-s − 4·21-s − 4·22-s + 3·24-s − 25-s + 26-s − 27-s − 4·28-s − 10·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.872·21-s − 0.852·22-s + 0.612·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20631 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20631 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.76585620841370, −15.28766404083386, −14.81666128085682, −14.28345145824823, −13.73218037577884, −13.02028576937107, −12.76854316648607, −12.03985806380365, −11.49020844727979, −11.09521942142235, −10.72658994658946, −9.798654995826896, −9.179817676641026, −8.403501344078620, −7.953147738690893, −7.574007993845584, −6.759963282232970, −5.652879086348631, −5.569625537251136, −4.817152098207844, −4.182358722571711, −3.956374869237469, −2.838689374136893, −2.061085853197599, −0.9044809538838308, 0,
0.9044809538838308, 2.061085853197599, 2.838689374136893, 3.956374869237469, 4.182358722571711, 4.817152098207844, 5.569625537251136, 5.652879086348631, 6.759963282232970, 7.574007993845584, 7.953147738690893, 8.403501344078620, 9.179817676641026, 9.798654995826896, 10.72658994658946, 11.09521942142235, 11.49020844727979, 12.03985806380365, 12.76854316648607, 13.02028576937107, 13.73218037577884, 14.28345145824823, 14.81666128085682, 15.28766404083386, 15.76585620841370
