| L(s) = 1 | − 5-s + 2·11-s + 6·13-s + 17-s + 8·19-s − 8·23-s + 25-s + 6·29-s + 8·31-s + 8·37-s − 6·41-s + 10·43-s − 8·47-s − 7·49-s + 14·53-s − 2·55-s − 6·61-s − 6·65-s − 2·67-s + 6·71-s − 4·73-s − 8·79-s + 4·83-s − 85-s + 8·89-s − 8·95-s + 12·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s + 1.66·13-s + 0.242·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 1.31·37-s − 0.937·41-s + 1.52·43-s − 1.16·47-s − 49-s + 1.92·53-s − 0.269·55-s − 0.768·61-s − 0.744·65-s − 0.244·67-s + 0.712·71-s − 0.468·73-s − 0.900·79-s + 0.439·83-s − 0.108·85-s + 0.847·89-s − 0.820·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589646134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589646134\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−16.27408422819681, −15.74542843315136, −15.50980268029461, −14.47390549148005, −14.10799441253504, −13.57403346646460, −13.06403948661963, −12.10572728444630, −11.72316509698590, −11.45788149655323, −10.49419280716993, −10.04711295470237, −9.349744395188395, −8.704632875327311, −8.007308127999438, −7.728470432273235, −6.675986914619620, −6.231677846328660, −5.609743800549409, −4.687597408756573, −3.994348846680126, −3.433694532439534, −2.665270702791336, −1.412637683006201, −0.8293525149789222,
0.8293525149789222, 1.412637683006201, 2.665270702791336, 3.433694532439534, 3.994348846680126, 4.687597408756573, 5.609743800549409, 6.231677846328660, 6.675986914619620, 7.728470432273235, 8.007308127999438, 8.704632875327311, 9.349744395188395, 10.04711295470237, 10.49419280716993, 11.45788149655323, 11.72316509698590, 12.10572728444630, 13.06403948661963, 13.57403346646460, 14.10799441253504, 14.47390549148005, 15.50980268029461, 15.74542843315136, 16.27408422819681
