| L(s) = 1 | + 5-s + 3·7-s + 4·11-s − 7·13-s + 4·17-s + 6·19-s + 6·23-s − 4·25-s − 4·29-s − 8·31-s + 3·35-s + 10·37-s + 8·41-s + 8·43-s − 3·47-s + 2·49-s − 2·53-s + 4·55-s + 59-s − 7·65-s + 4·67-s − 11·71-s − 6·73-s + 12·77-s + 79-s + 6·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 1.20·11-s − 1.94·13-s + 0.970·17-s + 1.37·19-s + 1.25·23-s − 4/5·25-s − 0.742·29-s − 1.43·31-s + 0.507·35-s + 1.64·37-s + 1.24·41-s + 1.21·43-s − 0.437·47-s + 2/7·49-s − 0.274·53-s + 0.539·55-s + 0.130·59-s − 0.868·65-s + 0.488·67-s − 1.30·71-s − 0.702·73-s + 1.36·77-s + 0.112·79-s + 0.658·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.081655921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.081655921\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−16.54364753434807, −16.05952113490001, −14.88907279924945, −14.61764021041996, −14.51601840318163, −13.74596944279568, −12.97881631400532, −12.36120249093336, −11.80775264636616, −11.34981837934827, −10.78199954759560, −9.777008144778638, −9.473201778950841, −9.094300653877485, −7.970456560965014, −7.409501663927649, −7.245336727119598, −6.037491846871241, −5.493032907253233, −4.900273228533534, −4.242505733007183, −3.324549022026335, −2.451052547669136, −1.650144237630809, −0.8621258959378183,
0.8621258959378183, 1.650144237630809, 2.451052547669136, 3.324549022026335, 4.242505733007183, 4.900273228533534, 5.493032907253233, 6.037491846871241, 7.245336727119598, 7.409501663927649, 7.970456560965014, 9.094300653877485, 9.473201778950841, 9.777008144778638, 10.78199954759560, 11.34981837934827, 11.80775264636616, 12.36120249093336, 12.97881631400532, 13.74596944279568, 14.51601840318163, 14.61764021041996, 14.88907279924945, 16.05952113490001, 16.54364753434807
