| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 21-s + 22-s + 24-s + 25-s + 27-s − 28-s + 6·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.449474167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.449474167\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 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
−12.32535198068856, −12.05947584556892, −11.83547805912749, −11.22037349145129, −10.62386898597389, −10.17351923193533, −9.752626470773796, −9.468771976891887, −8.648439487294068, −8.291363158070637, −7.959753966783822, −7.276409287058557, −7.001531078373138, −6.512487167457810, −5.939647066735248, −5.317046111218818, −5.061928045604840, −4.333684539366955, −3.880803009099189, −3.418576095675885, −2.941020673381445, −2.644451174656033, −1.699514425853478, −1.210560822592299, −0.5759783055798305,
0.5759783055798305, 1.210560822592299, 1.699514425853478, 2.644451174656033, 2.941020673381445, 3.418576095675885, 3.880803009099189, 4.333684539366955, 5.061928045604840, 5.317046111218818, 5.939647066735248, 6.512487167457810, 7.001531078373138, 7.276409287058557, 7.959753966783822, 8.291363158070637, 8.648439487294068, 9.468771976891887, 9.752626470773796, 10.17351923193533, 10.62386898597389, 11.22037349145129, 11.83547805912749, 12.05947584556892, 12.32535198068856
