| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 2·13-s + 16-s − 2·17-s − 18-s − 8·19-s − 4·23-s − 24-s − 2·26-s + 27-s − 2·29-s + 8·31-s − 32-s + 2·34-s + 36-s + 2·37-s + 8·38-s + 2·39-s − 6·41-s + 8·43-s + 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.83·19-s − 0.834·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + 1.29·38-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.91269776566401, −15.63085563336556, −15.08407701391907, −14.45148919352154, −13.99161858563000, −13.21858833380504, −12.86238274769050, −12.19430141246945, −11.47003850041842, −11.02093833566997, −10.25638670252749, −10.02936076105981, −9.138080076608209, −8.717425581308327, −8.204156328349199, −7.754365951431393, −6.872066942052640, −6.394496639684082, −5.866954201349874, −4.791318154762899, −4.138925812323313, −3.517871642617792, −2.494471301276727, −2.096095211751744, −1.118983332846726, 0,
1.118983332846726, 2.096095211751744, 2.494471301276727, 3.517871642617792, 4.138925812323313, 4.791318154762899, 5.866954201349874, 6.394496639684082, 6.872066942052640, 7.754365951431393, 8.204156328349199, 8.717425581308327, 9.138080076608209, 10.02936076105981, 10.25638670252749, 11.02093833566997, 11.47003850041842, 12.19430141246945, 12.86238274769050, 13.21858833380504, 13.99161858563000, 14.45148919352154, 15.08407701391907, 15.63085563336556, 15.91269776566401
