| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 4·11-s − 12-s − 13-s − 16-s + 6·17-s − 18-s − 8·19-s − 4·22-s + 4·23-s + 3·24-s + 26-s + 27-s − 2·29-s + 4·31-s − 5·32-s + 4·33-s − 6·34-s − 36-s − 10·37-s + 8·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.852·22-s + 0.834·23-s + 0.612·24-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.696·33-s − 1.02·34-s − 1/6·36-s − 1.64·37-s + 1.29·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.73831706479598, −14.31735748798843, −13.95762415477063, −13.32774613513231, −12.78100865152861, −12.30874017829872, −11.83650680034722, −10.97426460595015, −10.56637669068657, −10.01126741744129, −9.524674059162188, −9.063008940923401, −8.572788992596303, −8.150349882868349, −7.624329704012831, −6.818815825646607, −6.612003041737246, −5.600007103377036, −5.032853146539789, −4.328593958661545, −3.828728850139536, −3.282509492279062, −2.337422028597267, −1.573222038546134, −1.035613994855507, 0,
1.035613994855507, 1.573222038546134, 2.337422028597267, 3.282509492279062, 3.828728850139536, 4.328593958661545, 5.032853146539789, 5.600007103377036, 6.612003041737246, 6.818815825646607, 7.624329704012831, 8.150349882868349, 8.572788992596303, 9.063008940923401, 9.524674059162188, 10.01126741744129, 10.56637669068657, 10.97426460595015, 11.83650680034722, 12.30874017829872, 12.78100865152861, 13.32774613513231, 13.95762415477063, 14.31735748798843, 14.73831706479598
