| L(s) = 1 | + 3·5-s + 4·7-s − 2·11-s + 5·13-s − 6·17-s + 4·19-s + 4·25-s + 3·29-s − 6·31-s + 12·35-s − 6·37-s − 3·41-s + 2·43-s − 6·47-s + 9·49-s + 11·53-s − 6·55-s − 8·59-s − 15·61-s + 15·65-s − 16·67-s + 8·71-s − 13·73-s − 8·77-s − 4·79-s + 6·83-s − 18·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 0.603·11-s + 1.38·13-s − 1.45·17-s + 0.917·19-s + 4/5·25-s + 0.557·29-s − 1.07·31-s + 2.02·35-s − 0.986·37-s − 0.468·41-s + 0.304·43-s − 0.875·47-s + 9/7·49-s + 1.51·53-s − 0.809·55-s − 1.04·59-s − 1.92·61-s + 1.86·65-s − 1.95·67-s + 0.949·71-s − 1.52·73-s − 0.911·77-s − 0.450·79-s + 0.658·83-s − 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.60865890500119, −13.33695943994701, −12.75937075174940, −12.03613611530784, −11.60067538357697, −11.08989777768793, −10.56795577063679, −10.52546492606276, −9.711038602094208, −9.102029386085993, −8.739538212552064, −8.453002308357091, −7.651051352188738, −7.355027009988079, −6.543196683618357, −6.171563397629655, −5.510565045588205, −5.275343979349754, −4.620425041257228, −4.181260554016924, −3.318286047956703, −2.736519143694825, −1.973234161375889, −1.631794734950575, −1.169444624292312, 0,
1.169444624292312, 1.631794734950575, 1.973234161375889, 2.736519143694825, 3.318286047956703, 4.181260554016924, 4.620425041257228, 5.275343979349754, 5.510565045588205, 6.171563397629655, 6.543196683618357, 7.355027009988079, 7.651051352188738, 8.453002308357091, 8.739538212552064, 9.102029386085993, 9.711038602094208, 10.52546492606276, 10.56795577063679, 11.08989777768793, 11.60067538357697, 12.03613611530784, 12.75937075174940, 13.33695943994701, 13.60865890500119
