| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s − 11-s − 12-s + 13-s + 16-s − 5·17-s + 2·18-s + 3·19-s + 22-s + 4·23-s + 24-s − 26-s + 5·27-s + 29-s + 3·31-s − 32-s + 33-s + 5·34-s − 2·36-s + 9·37-s − 3·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 0.471·18-s + 0.688·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s + 0.962·27-s + 0.185·29-s + 0.538·31-s − 0.176·32-s + 0.174·33-s + 0.857·34-s − 1/3·36-s + 1.47·37-s − 0.486·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.76375788626197, −12.15096711971293, −11.62004195519422, −11.44106453417209, −10.87913989864108, −10.72205090229087, −10.01702417510052, −9.579175000174931, −9.155761402350677, −8.677992751504299, −8.095461233030484, −8.012671356366970, −7.067085257333190, −6.838681954687887, −6.353123090840385, −5.841496017334461, −5.325273000863071, −4.914602028072284, −4.311076382501057, −3.667103927432582, −2.968473186515235, −2.587875157915693, −2.038385346279020, −1.138859900934819, −0.7212055366100621, 0,
0.7212055366100621, 1.138859900934819, 2.038385346279020, 2.587875157915693, 2.968473186515235, 3.667103927432582, 4.311076382501057, 4.914602028072284, 5.325273000863071, 5.841496017334461, 6.353123090840385, 6.838681954687887, 7.067085257333190, 8.012671356366970, 8.095461233030484, 8.677992751504299, 9.155761402350677, 9.579175000174931, 10.01702417510052, 10.72205090229087, 10.87913989864108, 11.44106453417209, 11.62004195519422, 12.15096711971293, 12.76375788626197
