| L(s) = 1 | + 5·7-s + 13-s − 6·17-s − 5·19-s − 9·23-s + 4·31-s + 10·37-s − 3·41-s + 8·43-s − 3·47-s + 18·49-s − 3·53-s − 9·59-s + 8·61-s − 4·67-s − 6·71-s − 2·73-s − 2·79-s + 6·83-s + 6·89-s + 5·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 0.277·13-s − 1.45·17-s − 1.14·19-s − 1.87·23-s + 0.718·31-s + 1.64·37-s − 0.468·41-s + 1.21·43-s − 0.437·47-s + 18/7·49-s − 0.412·53-s − 1.17·59-s + 1.02·61-s − 0.488·67-s − 0.712·71-s − 0.234·73-s − 0.225·79-s + 0.658·83-s + 0.635·89-s + 0.524·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480926764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480926764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.81950884212750, −14.64244192365455, −14.05842188242099, −13.49504376481548, −13.06121087915139, −12.28170160915583, −11.72128191416915, −11.39730952552731, −10.71833914926088, −10.52544260044876, −9.651480369399073, −8.926968918120101, −8.525977708076715, −7.846757969169000, −7.759700819746638, −6.685364847387752, −6.208219406016186, −5.610656199769833, −4.761195618254557, −4.312444527209488, −4.044958212009499, −2.749388311334862, −2.065975597797337, −1.674448938359543, −0.5856060584573518,
0.5856060584573518, 1.674448938359543, 2.065975597797337, 2.749388311334862, 4.044958212009499, 4.312444527209488, 4.761195618254557, 5.610656199769833, 6.208219406016186, 6.685364847387752, 7.759700819746638, 7.846757969169000, 8.525977708076715, 8.926968918120101, 9.651480369399073, 10.52544260044876, 10.71833914926088, 11.39730952552731, 11.72128191416915, 12.28170160915583, 13.06121087915139, 13.49504376481548, 14.05842188242099, 14.64244192365455, 14.81950884212750
