| L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 4·19-s − 4·23-s + 25-s + 27-s + 2·29-s + 4·31-s + 4·33-s − 6·37-s − 2·39-s + 10·41-s + 8·43-s − 45-s − 7·49-s − 6·53-s − 4·55-s + 4·57-s + 8·59-s + 10·61-s + 2·65-s + 8·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s + 1.56·41-s + 1.21·43-s − 0.149·45-s − 49-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.04·59-s + 1.28·61-s + 0.248·65-s + 0.977·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.547748160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.547748160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−12.82252767899275, −12.17071358104890, −11.94469838991594, −11.41989505099439, −11.06559850633846, −10.32484703182789, −9.855572006500161, −9.605957011521249, −8.980539655980499, −8.658495960714285, −8.071274353691598, −7.622380781671819, −7.248143066146805, −6.656805961281982, −6.278566608729978, −5.612998782050416, −5.051521356427043, −4.454839215945834, −3.976299733425981, −3.637606670113351, −2.913789012975259, −2.494874976935674, −1.755078606966830, −1.144293215172832, −0.5256475556267163,
0.5256475556267163, 1.144293215172832, 1.755078606966830, 2.494874976935674, 2.913789012975259, 3.637606670113351, 3.976299733425981, 4.454839215945834, 5.051521356427043, 5.612998782050416, 6.278566608729978, 6.656805961281982, 7.248143066146805, 7.622380781671819, 8.071274353691598, 8.658495960714285, 8.980539655980499, 9.605957011521249, 9.855572006500161, 10.32484703182789, 11.06559850633846, 11.41989505099439, 11.94469838991594, 12.17071358104890, 12.82252767899275
