| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 4·7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 4·14-s + 15-s + 16-s + 2·17-s − 18-s − 4·19-s + 20-s − 4·21-s − 24-s + 25-s + 26-s + 27-s − 4·28-s + 2·29-s − 30-s − 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.182·30-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4643396864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4643396864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.01655466582118, −12.69105472191918, −12.15390517824766, −11.63877717961018, −11.04912809475492, −10.32290670074036, −10.13021008816899, −9.861562515024787, −9.110619896592144, −8.967879880856233, −8.458699459756667, −7.769879107990478, −7.393519826607020, −6.673198283232947, −6.529310758577092, −5.996154864232048, −5.280781977555029, −4.801384941066944, −3.895480837169112, −3.391252052038744, −3.105963204869915, −2.325282988228519, −1.902736800413604, −1.196316889221811, −0.2054412139957799,
0.2054412139957799, 1.196316889221811, 1.902736800413604, 2.325282988228519, 3.105963204869915, 3.391252052038744, 3.895480837169112, 4.801384941066944, 5.280781977555029, 5.996154864232048, 6.529310758577092, 6.673198283232947, 7.393519826607020, 7.769879107990478, 8.458699459756667, 8.967879880856233, 9.110619896592144, 9.861562515024787, 10.13021008816899, 10.32290670074036, 11.04912809475492, 11.63877717961018, 12.15390517824766, 12.69105472191918, 13.01655466582118
