| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 5·13-s − 2·14-s + 15-s + 16-s + 3·17-s − 18-s + 8·19-s + 20-s + 2·21-s + 22-s − 24-s + 25-s − 5·26-s + 27-s + 2·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.38·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.354628637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.354628637\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 89 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−15.15045985624388, −14.58434519457567, −14.16313226924397, −13.67732833876929, −13.12069424121931, −12.50702949862378, −11.88466560345574, −11.13124992401682, −10.95550403092889, −10.27168531927229, −9.581436674402538, −9.151357228681292, −8.783870966886846, −7.921226840783710, −7.647280244501996, −7.185537847926386, −6.255119986680167, −5.527499232728404, −5.354105360473623, −4.179831432111838, −3.560710135397515, −2.936288347463720, −2.083322734775502, −1.416250845535020, −0.8423998370086861,
0.8423998370086861, 1.416250845535020, 2.083322734775502, 2.936288347463720, 3.560710135397515, 4.179831432111838, 5.354105360473623, 5.527499232728404, 6.255119986680167, 7.185537847926386, 7.647280244501996, 7.921226840783710, 8.783870966886846, 9.151357228681292, 9.581436674402538, 10.27168531927229, 10.95550403092889, 11.13124992401682, 11.88466560345574, 12.50702949862378, 13.12069424121931, 13.67732833876929, 14.16313226924397, 14.58434519457567, 15.15045985624388
