| L(s) = 1 | − 3·9-s − 6·11-s + 2·13-s + 6·17-s − 6·19-s − 5·25-s − 6·29-s + 8·37-s − 6·41-s + 2·43-s + 8·47-s + 8·53-s − 4·59-s − 4·61-s − 2·67-s − 8·71-s − 6·73-s − 12·79-s + 9·81-s + 10·83-s + 10·89-s − 18·97-s + 18·99-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s − 1.37·19-s − 25-s − 1.11·29-s + 1.31·37-s − 0.937·41-s + 0.304·43-s + 1.16·47-s + 1.09·53-s − 0.520·59-s − 0.512·61-s − 0.244·67-s − 0.949·71-s − 0.702·73-s − 1.35·79-s + 81-s + 1.09·83-s + 1.05·89-s − 1.82·97-s + 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.44085826937462, −12.83611237304833, −12.29512233079321, −11.94243957609434, −11.26912704231973, −10.93295887644322, −10.50550228475536, −10.02161744705925, −9.629043592101368, −8.744553457208210, −8.653708730499396, −7.963775725998473, −7.612788552384403, −7.304887152281749, −6.274104784488141, −5.962268506869451, −5.560975951055682, −5.165092758231018, −4.406075319768771, −3.872679004494415, −3.244469830166145, −2.726844624253565, −2.256280926959024, −1.581885459179526, −0.6148674830492879, 0,
0.6148674830492879, 1.581885459179526, 2.256280926959024, 2.726844624253565, 3.244469830166145, 3.872679004494415, 4.406075319768771, 5.165092758231018, 5.560975951055682, 5.962268506869451, 6.274104784488141, 7.304887152281749, 7.612788552384403, 7.963775725998473, 8.653708730499396, 8.744553457208210, 9.629043592101368, 10.02161744705925, 10.50550228475536, 10.93295887644322, 11.26912704231973, 11.94243957609434, 12.29512233079321, 12.83611237304833, 13.44085826937462
