| L(s) = 1 | − 7-s + 6·13-s − 4·17-s − 19-s + 2·23-s − 5·25-s − 6·29-s − 3·31-s + 7·37-s − 2·41-s + 4·43-s + 6·47-s − 6·49-s + 2·53-s − 10·59-s − 61-s − 3·67-s + 4·71-s − 3·73-s + 5·79-s − 14·83-s − 16·89-s − 6·91-s + 13·97-s − 18·101-s + 15·103-s + 10·107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.66·13-s − 0.970·17-s − 0.229·19-s + 0.417·23-s − 25-s − 1.11·29-s − 0.538·31-s + 1.15·37-s − 0.312·41-s + 0.609·43-s + 0.875·47-s − 6/7·49-s + 0.274·53-s − 1.30·59-s − 0.128·61-s − 0.366·67-s + 0.474·71-s − 0.351·73-s + 0.562·79-s − 1.53·83-s − 1.69·89-s − 0.628·91-s + 1.31·97-s − 1.79·101-s + 1.47·103-s + 0.966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−7.43986637606515041714337942015, −6.64236425768975600880028051328, −6.04649822118206375147882469963, −5.55693473101826490306851355795, −4.43168100089311784548282105631, −3.90243485615828409780659228317, −3.15326372528734883995628220405, −2.16572902451137659191025904207, −1.27457842243263702345769041985, 0,
1.27457842243263702345769041985, 2.16572902451137659191025904207, 3.15326372528734883995628220405, 3.90243485615828409780659228317, 4.43168100089311784548282105631, 5.55693473101826490306851355795, 6.04649822118206375147882469963, 6.64236425768975600880028051328, 7.43986637606515041714337942015
