| L(s) = 1 | − 2·5-s + 4·7-s + 6·11-s − 2·17-s + 8·23-s − 25-s − 2·29-s + 8·31-s − 8·35-s − 8·37-s + 2·41-s − 8·43-s + 6·47-s + 9·49-s − 6·53-s − 12·55-s + 2·59-s + 2·61-s − 4·67-s + 6·71-s + 4·73-s + 24·77-s + 14·83-s + 4·85-s + 6·89-s − 12·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.80·11-s − 0.485·17-s + 1.66·23-s − 1/5·25-s − 0.371·29-s + 1.43·31-s − 1.35·35-s − 1.31·37-s + 0.312·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.61·55-s + 0.260·59-s + 0.256·61-s − 0.488·67-s + 0.712·71-s + 0.468·73-s + 2.73·77-s + 1.53·83-s + 0.433·85-s + 0.635·89-s − 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.888590275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888590275\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.29026783976082, −14.89638511448986, −14.45520901315998, −13.84239998222673, −13.44744107957503, −12.47430344162334, −11.96049147624409, −11.69999260452849, −11.02380123826957, −10.89264768390443, −9.857077530448533, −9.223219228799405, −8.612113943948468, −8.331685262868494, −7.593156251989166, −6.996523968637963, −6.570870258701931, −5.698375901753726, −4.780983261188023, −4.606978052228208, −3.804059380591402, −3.280919427033828, −2.147450695932024, −1.435562306869796, −0.7445975065021924,
0.7445975065021924, 1.435562306869796, 2.147450695932024, 3.280919427033828, 3.804059380591402, 4.606978052228208, 4.780983261188023, 5.698375901753726, 6.570870258701931, 6.996523968637963, 7.593156251989166, 8.331685262868494, 8.612113943948468, 9.223219228799405, 9.857077530448533, 10.89264768390443, 11.02380123826957, 11.69999260452849, 11.96049147624409, 12.47430344162334, 13.44744107957503, 13.84239998222673, 14.45520901315998, 14.89638511448986, 15.29026783976082
