| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 2·11-s + 12-s − 4·13-s + 16-s − 5·17-s − 18-s − 4·19-s + 2·22-s + 5·23-s − 24-s + 4·26-s + 27-s − 6·29-s − 11·31-s − 32-s − 2·33-s + 5·34-s + 36-s + 8·37-s + 4·38-s − 4·39-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 − 0.603·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.917·19-s + 0.426·22-s + 1.04·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 1.11·29-s − 1.97·31-s − 0.176·32-s − 0.348·33-s + 0.857·34-s + 1/6·36-s + 1.31·37-s + 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155735625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155735625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−7.86274562979873781100038969042, −7.33798766499990474972544031089, −6.84447839656809310984000229659, −5.89777984749581056236585265448, −5.07727140701149585704440551107, −4.28856601028708980310511363608, −3.39884071482364505378277622720, −2.31453756307238962736166452504, −2.10975701369475299698429777775, −0.55152933939773709777662125811,
0.55152933939773709777662125811, 2.10975701369475299698429777775, 2.31453756307238962736166452504, 3.39884071482364505378277622720, 4.28856601028708980310511363608, 5.07727140701149585704440551107, 5.89777984749581056236585265448, 6.84447839656809310984000229659, 7.33798766499990474972544031089, 7.86274562979873781100038969042
