| L(s) = 1 | − 5-s − 7-s + 13-s − 2·17-s + 4·19-s + 8·23-s + 25-s − 6·29-s − 4·31-s + 35-s + 6·37-s + 10·41-s + 12·47-s + 49-s − 2·53-s − 4·59-s − 10·61-s − 65-s + 8·67-s + 8·71-s + 6·73-s + 4·83-s + 2·85-s − 6·89-s − 91-s − 4·95-s + 6·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.169·35-s + 0.986·37-s + 1.56·41-s + 1.75·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.977·67-s + 0.949·71-s + 0.702·73-s + 0.439·83-s + 0.216·85-s − 0.635·89-s − 0.104·91-s − 0.410·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665225834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665225834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.72636158624074, −12.47082862915055, −11.91882982019099, −11.20095760817695, −11.04289981083362, −10.77085418480963, −9.965052356407812, −9.332825190447153, −9.244743991404159, −8.768251420422747, −8.012524602593961, −7.618122366756214, −7.189597911702113, −6.800555016946797, −6.081241970501201, −5.711758713257755, −5.144888412640520, −4.592285321213184, −4.032554090852144, −3.552290668045575, −2.995477217852656, −2.505130436009662, −1.776578154717281, −0.9469892251510286, −0.5421269514941223,
0.5421269514941223, 0.9469892251510286, 1.776578154717281, 2.505130436009662, 2.995477217852656, 3.552290668045575, 4.032554090852144, 4.592285321213184, 5.144888412640520, 5.711758713257755, 6.081241970501201, 6.800555016946797, 7.189597911702113, 7.618122366756214, 8.012524602593961, 8.768251420422747, 9.244743991404159, 9.332825190447153, 9.965052356407812, 10.77085418480963, 11.04289981083362, 11.20095760817695, 11.91882982019099, 12.47082862915055, 12.72636158624074
