| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 6·9-s − 6·12-s − 4·13-s − 4·16-s − 17-s − 12·18-s + 7·19-s − 7·23-s + 8·26-s − 9·27-s − 8·31-s + 8·32-s + 2·34-s + 12·36-s + 5·37-s − 14·38-s + 12·39-s − 8·41-s − 2·43-s + 14·46-s + 12·47-s + 12·48-s − 7·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 2·9-s − 1.73·12-s − 1.10·13-s − 16-s − 0.242·17-s − 2.82·18-s + 1.60·19-s − 1.45·23-s + 1.56·26-s − 1.73·27-s − 1.43·31-s + 1.41·32-s + 0.342·34-s + 2·36-s + 0.821·37-s − 2.27·38-s + 1.92·39-s − 1.24·41-s − 0.304·43-s + 2.06·46-s + 1.75·47-s + 1.73·48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1890890829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1890890829\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−14.49768944140995, −14.04865597159935, −13.23390190859719, −12.83987359727190, −12.01461545054650, −11.80660189022320, −11.40409097253206, −10.79300483532061, −10.23173078769586, −9.942488899585779, −9.507028898333869, −8.903588332428662, −8.101625055688053, −7.560187157538569, −7.124913150680831, −6.765350914497873, −5.869651726477466, −5.544115804244346, −4.869242252702580, −4.357512526034251, −3.539733960677102, −2.417735534040758, −1.739274422655507, −1.020136475854815, −0.2612475301562948,
0.2612475301562948, 1.020136475854815, 1.739274422655507, 2.417735534040758, 3.539733960677102, 4.357512526034251, 4.869242252702580, 5.544115804244346, 5.869651726477466, 6.765350914497873, 7.124913150680831, 7.560187157538569, 8.101625055688053, 8.903588332428662, 9.507028898333869, 9.942488899585779, 10.23173078769586, 10.79300483532061, 11.40409097253206, 11.80660189022320, 12.01461545054650, 12.83987359727190, 13.23390190859719, 14.04865597159935, 14.49768944140995
