| L(s) = 1 | + 5-s + 7-s + 4·13-s + 6·17-s + 4·19-s − 23-s − 4·25-s + 2·29-s + 31-s + 35-s − 9·41-s − 43-s + 4·47-s − 6·49-s − 3·53-s − 14·59-s − 2·61-s + 4·65-s − 10·67-s − 2·71-s + 79-s − 12·83-s + 6·85-s − 9·89-s + 4·91-s + 4·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 0.208·23-s − 4/5·25-s + 0.371·29-s + 0.179·31-s + 0.169·35-s − 1.40·41-s − 0.152·43-s + 0.583·47-s − 6/7·49-s − 0.412·53-s − 1.82·59-s − 0.256·61-s + 0.496·65-s − 1.22·67-s − 0.237·71-s + 0.112·79-s − 1.31·83-s + 0.650·85-s − 0.953·89-s + 0.419·91-s + 0.410·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.46233594759948, −12.80749808693050, −12.34273304915065, −11.76249249261308, −11.59730330565405, −10.91071473115380, −10.46866318788078, −9.900385681167969, −9.706279022657202, −9.026438617562831, −8.505605142112078, −8.105353025799178, −7.548705988888743, −7.215062694208348, −6.371236153338847, −5.997245235976679, −5.637370128142803, −5.012526291872186, −4.543179245161562, −3.812403085012671, −3.247042768652817, −2.965532552029545, −1.948936402527584, −1.468012844902843, −1.034236323501784, 0,
1.034236323501784, 1.468012844902843, 1.948936402527584, 2.965532552029545, 3.247042768652817, 3.812403085012671, 4.543179245161562, 5.012526291872186, 5.637370128142803, 5.997245235976679, 6.371236153338847, 7.215062694208348, 7.548705988888743, 8.105353025799178, 8.505605142112078, 9.026438617562831, 9.706279022657202, 9.900385681167969, 10.46866318788078, 10.91071473115380, 11.59730330565405, 11.76249249261308, 12.34273304915065, 12.80749808693050, 13.46233594759948
