| L(s) = 1 | + 2-s + 3·3-s + 4-s + 3·6-s + 8-s + 6·9-s − 3·11-s + 3·12-s + 5·13-s + 16-s + 5·17-s + 6·18-s − 3·22-s − 8·23-s + 3·24-s + 5·26-s + 9·27-s − 2·29-s − 31-s + 32-s − 9·33-s + 5·34-s + 6·36-s − 7·37-s + 15·39-s + 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.904·11-s + 0.866·12-s + 1.38·13-s + 1/4·16-s + 1.21·17-s + 1.41·18-s − 0.639·22-s − 1.66·23-s + 0.612·24-s + 0.980·26-s + 1.73·27-s − 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.56·33-s + 0.857·34-s + 36-s − 1.15·37-s + 2.40·39-s + 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.33451793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.33451793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.29570987700177, −12.80361602611166, −12.36701891722665, −11.78515134989406, −11.34903124689161, −10.42606047231576, −10.32388554041565, −9.941913173266960, −9.061198922393151, −8.842491494393776, −8.244446537311180, −7.802429139653981, −7.570722513992118, −6.943230434573050, −6.260824823875722, −5.711557906276542, −5.332590629015362, −4.517842974265814, −3.894660065055531, −3.670397010730285, −3.136419671017563, −2.618276966837642, −1.938973493725222, −1.598674217784864, −0.6826163133432745,
0.6826163133432745, 1.598674217784864, 1.938973493725222, 2.618276966837642, 3.136419671017563, 3.670397010730285, 3.894660065055531, 4.517842974265814, 5.332590629015362, 5.711557906276542, 6.260824823875722, 6.943230434573050, 7.570722513992118, 7.802429139653981, 8.244446537311180, 8.842491494393776, 9.061198922393151, 9.941913173266960, 10.32388554041565, 10.42606047231576, 11.34903124689161, 11.78515134989406, 12.36701891722665, 12.80361602611166, 13.29570987700177
