| L(s) = 1 | + 4·5-s + 2·17-s + 4·23-s + 11·25-s − 2·29-s − 8·31-s + 8·37-s − 4·41-s + 8·43-s − 6·53-s − 8·59-s + 10·61-s + 8·67-s + 4·73-s + 12·79-s − 8·83-s + 8·85-s − 12·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 16·115-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.485·17-s + 0.834·23-s + 11/5·25-s − 0.371·29-s − 1.43·31-s + 1.31·37-s − 0.624·41-s + 1.21·43-s − 0.824·53-s − 1.04·59-s + 1.28·61-s + 0.977·67-s + 0.468·73-s + 1.35·79-s − 0.878·83-s + 0.867·85-s − 1.27·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.49·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.97148063518627, −12.57330020419760, −12.22858098532548, −11.28852160997116, −11.09924633911249, −10.63541604384428, −10.08645907149223, −9.632250576683762, −9.311262655570275, −9.034476047176209, −8.325447925022169, −7.837156257458598, −7.236710204765265, −6.736202172653076, −6.346351248426136, −5.741197015643282, −5.425094856777660, −5.065160938054357, −4.363145687031351, −3.725832224065044, −3.103909279537950, −2.493748883578274, −2.165075199523793, −1.347450890822948, −1.097353294209362, 0,
1.097353294209362, 1.347450890822948, 2.165075199523793, 2.493748883578274, 3.103909279537950, 3.725832224065044, 4.363145687031351, 5.065160938054357, 5.425094856777660, 5.741197015643282, 6.346351248426136, 6.736202172653076, 7.236710204765265, 7.837156257458598, 8.325447925022169, 9.034476047176209, 9.311262655570275, 9.632250576683762, 10.08645907149223, 10.63541604384428, 11.09924633911249, 11.28852160997116, 12.22858098532548, 12.57330020419760, 12.97148063518627
