| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 7-s + 8-s + 9-s − 2·10-s − 2·11-s − 12-s − 6·13-s + 14-s + 2·15-s + 16-s + 2·17-s + 18-s − 2·20-s − 21-s − 2·22-s − 23-s − 24-s − 25-s − 6·26-s − 27-s + 28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.218·21-s − 0.426·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954950765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954950765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.32449928543436, −12.19265778115671, −11.86902234217587, −11.18498857279086, −10.87555636251491, −10.51749089077470, −9.930137265816703, −9.445672496774493, −8.957595284377276, −8.197172816666080, −7.647888873416542, −7.542454242515189, −7.108056338017001, −6.526346434517065, −5.693382702502717, −5.593156779335417, −4.940205342829537, −4.636086264452788, −3.993583396067324, −3.664044484107486, −2.961533413762733, −2.238510683426608, −2.039918157324580, −0.9365756527478571, −0.3876645373207839,
0.3876645373207839, 0.9365756527478571, 2.039918157324580, 2.238510683426608, 2.961533413762733, 3.664044484107486, 3.993583396067324, 4.636086264452788, 4.940205342829537, 5.593156779335417, 5.693382702502717, 6.526346434517065, 7.108056338017001, 7.542454242515189, 7.647888873416542, 8.197172816666080, 8.957595284377276, 9.445672496774493, 9.930137265816703, 10.51749089077470, 10.87555636251491, 11.18498857279086, 11.86902234217587, 12.19265778115671, 12.32449928543436
