| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s − 2·9-s + 3·11-s − 12-s − 6·13-s + 4·14-s + 16-s − 3·17-s − 2·18-s + 3·19-s − 4·21-s + 3·22-s − 2·23-s − 24-s − 6·26-s + 5·27-s + 4·28-s + 32-s − 3·33-s − 3·34-s − 2·36-s + 3·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.688·19-s − 0.872·21-s + 0.639·22-s − 0.417·23-s − 0.204·24-s − 1.17·26-s + 0.962·27-s + 0.755·28-s + 0.176·32-s − 0.522·33-s − 0.514·34-s − 1/3·36-s + 0.486·38-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.35699894777368, −14.13667352693145, −13.54124351049825, −12.84090053178397, −12.14177951413174, −11.86228540204481, −11.67155947247978, −10.98375633239750, −10.68328419408988, −9.949315788208852, −9.318332319038291, −8.811348533933173, −8.063367701538761, −7.753566866681740, −6.936666471229140, −6.732632016914127, −5.793337070548702, −5.429900213142230, −4.948781980991746, −4.414084702107994, −3.989721222841359, −2.978086048329551, −2.430136992320514, −1.782569910008573, −1.038909867614247, 0,
1.038909867614247, 1.782569910008573, 2.430136992320514, 2.978086048329551, 3.989721222841359, 4.414084702107994, 4.948781980991746, 5.429900213142230, 5.793337070548702, 6.732632016914127, 6.936666471229140, 7.753566866681740, 8.063367701538761, 8.811348533933173, 9.318332319038291, 9.949315788208852, 10.68328419408988, 10.98375633239750, 11.67155947247978, 11.86228540204481, 12.14177951413174, 12.84090053178397, 13.54124351049825, 14.13667352693145, 14.35699894777368
