| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s + 8-s + 6·9-s − 3·11-s + 3·12-s + 13-s + 16-s − 7·17-s + 6·18-s + 19-s − 3·22-s − 4·23-s + 3·24-s + 26-s + 9·27-s + 4·29-s − 10·31-s + 32-s − 9·33-s − 7·34-s + 6·36-s + 12·37-s + 38-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.904·11-s + 0.866·12-s + 0.277·13-s + 1/4·16-s − 1.69·17-s + 1.41·18-s + 0.229·19-s − 0.639·22-s − 0.834·23-s + 0.612·24-s + 0.196·26-s + 1.73·27-s + 0.742·29-s − 1.79·31-s + 0.176·32-s − 1.56·33-s − 1.20·34-s + 36-s + 1.97·37-s + 0.162·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.717116588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.717116588\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 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
−10.51863568275579508375303678319, −9.534079111068019193968555221014, −8.744258785488799933121780172107, −7.921987388760171906149019453046, −7.21774047311328451412817428115, −6.07796556325317003523478387578, −4.68808727774989385862743878780, −3.86179772916415191818436864225, −2.79015867289541412533528864387, −2.01747373249326413385058901434,
2.01747373249326413385058901434, 2.79015867289541412533528864387, 3.86179772916415191818436864225, 4.68808727774989385862743878780, 6.07796556325317003523478387578, 7.21774047311328451412817428115, 7.921987388760171906149019453046, 8.744258785488799933121780172107, 9.534079111068019193968555221014, 10.51863568275579508375303678319
