| L(s) = 1 | − 3-s + 4·7-s − 2·9-s − 3·11-s + 2·13-s − 6·17-s − 4·21-s + 6·23-s − 5·25-s + 5·27-s − 2·31-s + 3·33-s − 10·37-s − 2·39-s + 9·41-s + 4·43-s + 9·49-s + 6·51-s + 6·53-s + 9·59-s − 4·61-s − 8·63-s + 7·67-s − 6·69-s + 6·71-s − 73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.904·11-s + 0.554·13-s − 1.45·17-s − 0.872·21-s + 1.25·23-s − 25-s + 0.962·27-s − 0.359·31-s + 0.522·33-s − 1.64·37-s − 0.320·39-s + 1.40·41-s + 0.609·43-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 1.17·59-s − 0.512·61-s − 1.00·63-s + 0.855·67-s − 0.722·69-s + 0.712·71-s − 0.117·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.468907456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468907456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−8.172579614354324377311670896408, −7.44101697136447859157294105288, −6.69372501572221659699868927176, −5.77808754119992068778159446976, −5.24384583995636037802031925040, −4.68831140093909709494335669481, −3.80517416710362653522402828678, −2.61087972354877885021627800415, −1.89014609885089839433061628082, −0.65470027547235903230677355928,
0.65470027547235903230677355928, 1.89014609885089839433061628082, 2.61087972354877885021627800415, 3.80517416710362653522402828678, 4.68831140093909709494335669481, 5.24384583995636037802031925040, 5.77808754119992068778159446976, 6.69372501572221659699868927176, 7.44101697136447859157294105288, 8.172579614354324377311670896408
