| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 12-s + 4·13-s + 16-s + 2·18-s − 4·19-s − 24-s − 4·26-s − 5·27-s + 6·29-s + 10·31-s − 32-s − 2·36-s − 8·37-s + 4·38-s + 4·39-s − 3·41-s − 43-s + 9·47-s + 48-s + 4·52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.471·18-s − 0.917·19-s − 0.204·24-s − 0.784·26-s − 0.962·27-s + 1.11·29-s + 1.79·31-s − 0.176·32-s − 1/3·36-s − 1.31·37-s + 0.648·38-s + 0.640·39-s − 0.468·41-s − 0.152·43-s + 1.31·47-s + 0.144·48-s + 0.554·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.13701973470334, −12.19890904562506, −11.98300935003337, −11.61116972962998, −10.94429487466922, −10.52542344817960, −10.25164864469290, −9.718495149098000, −8.965287547801487, −8.742354167095905, −8.429126656272599, −8.075411483145131, −7.444283040756432, −6.829571429917763, −6.492762339392068, −5.900905806529698, −5.568904144045015, −4.746792273140190, −4.251348185938442, −3.564998592447368, −3.204272093578526, −2.458741970314780, −2.221971814458745, −1.333896673323824, −0.8268302343100176, 0,
0.8268302343100176, 1.333896673323824, 2.221971814458745, 2.458741970314780, 3.204272093578526, 3.564998592447368, 4.251348185938442, 4.746792273140190, 5.568904144045015, 5.900905806529698, 6.492762339392068, 6.829571429917763, 7.444283040756432, 8.075411483145131, 8.429126656272599, 8.742354167095905, 8.965287547801487, 9.718495149098000, 10.25164864469290, 10.52542344817960, 10.94429487466922, 11.61116972962998, 11.98300935003337, 12.19890904562506, 13.13701973470334
