| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s − 11-s − 2·13-s + 16-s + 2·19-s − 2·20-s + 22-s − 25-s + 2·26-s − 6·29-s − 4·31-s − 32-s + 2·37-s − 2·38-s + 2·40-s + 8·41-s + 12·43-s − 44-s + 12·47-s + 50-s − 2·52-s + 2·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.458·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s + 0.392·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.328·37-s − 0.324·38-s + 0.316·40-s + 1.24·41-s + 1.82·43-s − 0.150·44-s + 1.75·47-s + 0.141·50-s − 0.277·52-s + 0.274·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.37879324142005899641967830967, −7.12718699626659355908408781390, −5.85681260756016898432318007880, −5.54145784016091300416607843536, −4.34837378838810542722999730423, −3.88039108882371738271699362297, −2.86920671461020559917351265420, −2.18047062847661088369700678833, −0.978990914696866592185433823971, 0,
0.978990914696866592185433823971, 2.18047062847661088369700678833, 2.86920671461020559917351265420, 3.88039108882371738271699362297, 4.34837378838810542722999730423, 5.54145784016091300416607843536, 5.85681260756016898432318007880, 7.12718699626659355908408781390, 7.37879324142005899641967830967
