| L(s) = 1 | − 5-s − 3·7-s + 3·11-s + 4·13-s + 19-s + 2·23-s + 25-s + 5·29-s + 4·31-s + 3·35-s − 5·37-s + 5·41-s − 2·43-s + 3·47-s + 2·49-s + 9·53-s − 3·55-s − 4·59-s − 6·61-s − 4·65-s + 14·67-s + 8·71-s − 11·73-s − 9·77-s − 12·79-s + 4·83-s + 12·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.904·11-s + 1.10·13-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 0.928·29-s + 0.718·31-s + 0.507·35-s − 0.821·37-s + 0.780·41-s − 0.304·43-s + 0.437·47-s + 2/7·49-s + 1.23·53-s − 0.404·55-s − 0.520·59-s − 0.768·61-s − 0.496·65-s + 1.71·67-s + 0.949·71-s − 1.28·73-s − 1.02·77-s − 1.35·79-s + 0.439·83-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.23174764491935, −12.76760438057842, −12.34662844273879, −11.77768034971600, −11.58288448162258, −10.81035814191328, −10.56215509614756, −9.901599779384720, −9.519450724740815, −8.971880621082666, −8.565736228866737, −8.206899544397020, −7.366460712494174, −7.055659844401003, −6.427085955717080, −6.207154111315928, −5.643633352117625, −4.879170581731499, −4.384428163939062, −3.663586626078776, −3.524526828079582, −2.857258889103047, −2.233474662013498, −1.231965334464109, −0.9137242620952247, 0,
0.9137242620952247, 1.231965334464109, 2.233474662013498, 2.857258889103047, 3.524526828079582, 3.663586626078776, 4.384428163939062, 4.879170581731499, 5.643633352117625, 6.207154111315928, 6.427085955717080, 7.055659844401003, 7.366460712494174, 8.206899544397020, 8.565736228866737, 8.971880621082666, 9.519450724740815, 9.901599779384720, 10.56215509614756, 10.81035814191328, 11.58288448162258, 11.77768034971600, 12.34662844273879, 12.76760438057842, 13.23174764491935
