| L(s) = 1 | + 4·5-s − 7-s + 3·11-s − 5·13-s − 7·17-s + 5·19-s − 23-s + 11·25-s + 8·29-s + 6·31-s − 4·35-s + 37-s + 2·41-s + 12·43-s + 2·47-s − 6·49-s + 53-s + 12·55-s + 4·59-s + 14·61-s − 20·65-s + 14·67-s − 4·71-s − 11·73-s − 3·77-s + 10·79-s − 5·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s + 0.904·11-s − 1.38·13-s − 1.69·17-s + 1.14·19-s − 0.208·23-s + 11/5·25-s + 1.48·29-s + 1.07·31-s − 0.676·35-s + 0.164·37-s + 0.312·41-s + 1.82·43-s + 0.291·47-s − 6/7·49-s + 0.137·53-s + 1.61·55-s + 0.520·59-s + 1.79·61-s − 2.48·65-s + 1.71·67-s − 0.474·71-s − 1.28·73-s − 0.341·77-s + 1.12·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.515638803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.515638803\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−9.170919281527884387381532166892, −8.238413206921816077708675930179, −6.93103230973337401103812300140, −6.65969419424467214117062769830, −5.81739413469973915405265897976, −5.02471432618468510430501908595, −4.23965014562456207446590174902, −2.74420683922802619411355240930, −2.28704150430081549709622153078, −1.03487491472005502033801272705,
1.03487491472005502033801272705, 2.28704150430081549709622153078, 2.74420683922802619411355240930, 4.23965014562456207446590174902, 5.02471432618468510430501908595, 5.81739413469973915405265897976, 6.65969419424467214117062769830, 6.93103230973337401103812300140, 8.238413206921816077708675930179, 9.170919281527884387381532166892
