| L(s) = 1 | + 3-s − 5-s − 2·9-s − 3·11-s − 2·13-s − 15-s + 3·17-s − 19-s − 6·23-s + 25-s − 5·27-s − 6·29-s + 5·31-s − 3·33-s − 7·37-s − 2·39-s + 6·41-s + 43-s + 2·45-s + 3·47-s + 3·51-s − 9·53-s + 3·55-s − 57-s + 6·59-s − 8·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 0.554·13-s − 0.258·15-s + 0.727·17-s − 0.229·19-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 1.11·29-s + 0.898·31-s − 0.522·33-s − 1.15·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.298·45-s + 0.437·47-s + 0.420·51-s − 1.23·53-s + 0.404·55-s − 0.132·57-s + 0.781·59-s − 1.02·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.75180302988474, −12.51688491778347, −12.00123382904366, −11.49557420533166, −11.15469257674529, −10.53933099578769, −10.09250542163724, −9.766330711801862, −9.070822408342892, −8.798827226485166, −8.029441444381010, −7.959502598995963, −7.502608273765842, −6.986116251162065, −6.215068079845518, −5.827032133594971, −5.349661791591284, −4.800129949024821, −4.253721453447561, −3.590972061483674, −3.280308227069901, −2.592018847333583, −2.228390644623857, −1.582854368855556, −0.5884792905777664, 0,
0.5884792905777664, 1.582854368855556, 2.228390644623857, 2.592018847333583, 3.280308227069901, 3.590972061483674, 4.253721453447561, 4.800129949024821, 5.349661791591284, 5.827032133594971, 6.215068079845518, 6.986116251162065, 7.502608273765842, 7.959502598995963, 8.029441444381010, 8.798827226485166, 9.070822408342892, 9.766330711801862, 10.09250542163724, 10.53933099578769, 11.15469257674529, 11.49557420533166, 12.00123382904366, 12.51688491778347, 12.75180302988474
