| L(s) = 1 | − 5-s + 6·11-s + 13-s − 2·17-s + 2·19-s + 4·23-s + 25-s − 2·29-s + 2·31-s + 2·37-s − 6·41-s + 6·47-s + 2·53-s − 6·55-s − 6·59-s − 14·61-s − 65-s + 2·67-s + 10·71-s + 6·73-s + 4·79-s + 2·83-s + 2·85-s − 14·89-s − 2·95-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·11-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.359·31-s + 0.328·37-s − 0.937·41-s + 0.875·47-s + 0.274·53-s − 0.809·55-s − 0.781·59-s − 1.79·61-s − 0.124·65-s + 0.244·67-s + 1.18·71-s + 0.702·73-s + 0.450·79-s + 0.219·83-s + 0.216·85-s − 1.48·89-s − 0.205·95-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.85290274088981, −13.54066950302156, −12.76877496834823, −12.32894225142594, −11.92983048984970, −11.43603291559627, −11.01503985716777, −10.62918817135074, −9.747912127012169, −9.461958650739369, −8.895705225602286, −8.589623768075742, −7.887470316784162, −7.368346495879638, −6.738675472464090, −6.524306891515224, −5.854527061370481, −5.200107520063036, −4.572591885890738, −4.071280910839130, −3.588783175443896, −3.033702369964955, −2.258768369119348, −1.391783731098836, −1.031621882934532, 0,
1.031621882934532, 1.391783731098836, 2.258768369119348, 3.033702369964955, 3.588783175443896, 4.071280910839130, 4.572591885890738, 5.200107520063036, 5.854527061370481, 6.524306891515224, 6.738675472464090, 7.368346495879638, 7.887470316784162, 8.589623768075742, 8.895705225602286, 9.461958650739369, 9.747912127012169, 10.62918817135074, 11.01503985716777, 11.43603291559627, 11.92983048984970, 12.32894225142594, 12.76877496834823, 13.54066950302156, 13.85290274088981
