| L(s) = 1 | + 5-s + 4·7-s + 4·11-s + 13-s + 6·17-s − 8·23-s + 25-s − 6·29-s + 4·31-s + 4·35-s − 2·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·53-s + 4·55-s + 12·59-s − 2·61-s + 65-s + 16·67-s − 8·71-s − 6·73-s + 16·77-s + 16·79-s + 4·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.20·11-s + 0.277·13-s + 1.45·17-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.124·65-s + 1.95·67-s − 0.949·71-s − 0.702·73-s + 1.82·77-s + 1.80·79-s + 0.439·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.457450983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457450983\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.968682759424561933831757091255, −7.07262364728116113208537629201, −6.21672147498484835051363161803, −5.65968463169530375708945420358, −5.03104716929566360549516787498, −4.10925011496621641941230567060, −3.67896218613553789648435530758, −2.40292444726231390025877377076, −1.64504520317014171091379354867, −0.992568785942682534383301571568,
0.992568785942682534383301571568, 1.64504520317014171091379354867, 2.40292444726231390025877377076, 3.67896218613553789648435530758, 4.10925011496621641941230567060, 5.03104716929566360549516787498, 5.65968463169530375708945420358, 6.21672147498484835051363161803, 7.07262364728116113208537629201, 7.968682759424561933831757091255
