| L(s) = 1 | + 3-s − 5-s + 3·7-s + 9-s + 4·11-s + 3·13-s − 15-s + 7·19-s + 3·21-s + 4·23-s + 25-s + 27-s − 29-s − 3·31-s + 4·33-s − 3·35-s − 12·37-s + 3·39-s + 9·41-s + 43-s − 45-s − 6·47-s + 2·49-s + 6·53-s − 4·55-s + 7·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s + 0.832·13-s − 0.258·15-s + 1.60·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s − 0.538·31-s + 0.696·33-s − 0.507·35-s − 1.97·37-s + 0.480·39-s + 1.40·41-s + 0.152·43-s − 0.149·45-s − 0.875·47-s + 2/7·49-s + 0.824·53-s − 0.539·55-s + 0.927·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.589089610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.589089610\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−14.50668202519520, −14.28468844460460, −13.95925085019776, −13.28411691509392, −12.64751338856903, −12.09421574449721, −11.50916341410409, −11.22625499674335, −10.71787739686319, −9.898944421729776, −9.352557177450574, −8.758538584651194, −8.541340051050090, −7.734870298858769, −7.350843176614181, −6.831571461284986, −6.059659673111826, −5.319626853671727, −4.827480996516524, −4.101367395354299, −3.539392035192407, −3.105032779347094, −2.013152799794646, −1.406013130020736, −0.8397690829003723,
0.8397690829003723, 1.406013130020736, 2.013152799794646, 3.105032779347094, 3.539392035192407, 4.101367395354299, 4.827480996516524, 5.319626853671727, 6.059659673111826, 6.831571461284986, 7.350843176614181, 7.734870298858769, 8.541340051050090, 8.758538584651194, 9.352557177450574, 9.898944421729776, 10.71787739686319, 11.22625499674335, 11.50916341410409, 12.09421574449721, 12.64751338856903, 13.28411691509392, 13.95925085019776, 14.28468844460460, 14.50668202519520
