| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s − 11-s − 2·13-s + 14-s + 16-s + 4·17-s − 2·19-s − 2·20-s + 22-s − 25-s + 2·26-s − 28-s − 29-s − 9·31-s − 32-s − 4·34-s + 2·35-s + 2·37-s + 2·38-s + 2·40-s + 6·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.185·29-s − 1.61·31-s − 0.176·32-s − 0.685·34-s + 0.338·35-s + 0.328·37-s + 0.324·38-s + 0.316·40-s + 0.937·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5423177061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5423177061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−7.71103709837817316661949447511, −7.27613924138377091001188242080, −6.49654155842629661623143784889, −5.71805540722915589014498621055, −4.99932062179299903886756330539, −4.01714355260089192782945885909, −3.41147328594117735176342256862, −2.58376225214808313422379671204, −1.60744029620020824663885869515, −0.38759347731831070655166421320,
0.38759347731831070655166421320, 1.60744029620020824663885869515, 2.58376225214808313422379671204, 3.41147328594117735176342256862, 4.01714355260089192782945885909, 4.99932062179299903886756330539, 5.71805540722915589014498621055, 6.49654155842629661623143784889, 7.27613924138377091001188242080, 7.71103709837817316661949447511
