| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s + 4·11-s − 2·12-s − 2·13-s + 16-s + 8·17-s + 18-s + 5·19-s + 4·22-s + 23-s − 2·24-s − 2·26-s + 4·27-s − 10·29-s + 4·31-s + 32-s − 8·33-s + 8·34-s + 36-s + 5·38-s + 4·39-s + 7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 1.14·19-s + 0.852·22-s + 0.208·23-s − 0.408·24-s − 0.392·26-s + 0.769·27-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.39·33-s + 1.37·34-s + 1/6·36-s + 0.811·38-s + 0.640·39-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.146692847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.146692847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 11 T + p T^{2} \) | 1.59.al |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.15347345034897, −13.79438864893811, −12.95291378901957, −12.60714225813657, −12.06069288878946, −11.63444342313720, −11.45495643119303, −10.88923599015718, −10.03397990115994, −9.819386706415838, −9.291747878872213, −8.429202235682557, −7.782784947810370, −7.289382216595305, −6.761739977020309, −6.183177005451836, −5.696615065947870, −5.223066548875396, −4.873715531381008, −4.018988145170201, −3.447060445557841, −3.010367973852332, −1.947334535543526, −1.230348723102678, −0.6286509248166450,
0.6286509248166450, 1.230348723102678, 1.947334535543526, 3.010367973852332, 3.447060445557841, 4.018988145170201, 4.873715531381008, 5.223066548875396, 5.696615065947870, 6.183177005451836, 6.761739977020309, 7.289382216595305, 7.782784947810370, 8.429202235682557, 9.291747878872213, 9.819386706415838, 10.03397990115994, 10.88923599015718, 11.45495643119303, 11.63444342313720, 12.06069288878946, 12.60714225813657, 12.95291378901957, 13.79438864893811, 14.15347345034897
