| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 11-s + 2·14-s − 16-s − 2·17-s + 8·19-s − 22-s + 6·23-s + 2·28-s + 10·29-s + 4·31-s − 5·32-s + 2·34-s − 6·37-s − 8·38-s + 6·41-s − 10·43-s − 44-s − 6·46-s − 6·47-s − 3·49-s − 6·53-s − 6·56-s − 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 0.301·11-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.213·22-s + 1.25·23-s + 0.377·28-s + 1.85·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.986·37-s − 1.29·38-s + 0.937·41-s − 1.52·43-s − 0.150·44-s − 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 0.801·56-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9378907124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9378907124\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.35934960393767, −12.78174554623512, −12.25493801557938, −11.90029160679101, −11.13510769257813, −10.91210932492274, −10.12845321914871, −9.739918520520941, −9.626732454450589, −8.837627049171506, −8.620123804660612, −8.064446036617906, −7.388465334307273, −7.075718538879166, −6.471372404576620, −6.023124110860962, −5.164117845156744, −4.811557779207809, −4.424814557817649, −3.417093792266995, −3.229208036266096, −2.582746211274982, −1.457707109222283, −1.193174584341688, −0.3609167381848429,
0.3609167381848429, 1.193174584341688, 1.457707109222283, 2.582746211274982, 3.229208036266096, 3.417093792266995, 4.424814557817649, 4.811557779207809, 5.164117845156744, 6.023124110860962, 6.471372404576620, 7.075718538879166, 7.388465334307273, 8.064446036617906, 8.620123804660612, 8.837627049171506, 9.626732454450589, 9.739918520520941, 10.12845321914871, 10.91210932492274, 11.13510769257813, 11.90029160679101, 12.25493801557938, 12.78174554623512, 13.35934960393767
