| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 5·11-s − 13-s + 15-s + 6·19-s − 21-s + 6·23-s − 4·25-s − 27-s − 6·29-s + 4·31-s − 5·33-s − 35-s − 11·37-s + 39-s + 9·43-s − 45-s − 4·47-s + 49-s − 7·53-s − 5·55-s − 6·57-s − 12·59-s − 6·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s + 1.37·19-s − 0.218·21-s + 1.25·23-s − 4/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.870·33-s − 0.169·35-s − 1.80·37-s + 0.160·39-s + 1.37·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s − 0.961·53-s − 0.674·55-s − 0.794·57-s − 1.56·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.01536861185948, −13.71250315329256, −12.94067224905971, −12.36665667490016, −11.99823917389867, −11.65985672070139, −11.10634047735888, −10.83734602766199, −10.05244414763490, −9.509967978515312, −9.117652890009725, −8.706680016658911, −7.768362503134963, −7.505592141977367, −7.056959246354084, −6.302444861804662, −6.026001920027076, −5.137625289439302, −4.883737290507347, −4.214706703453016, −3.505164024380132, −3.252815928229884, −2.147312047453889, −1.447539350583245, −0.9586229266453453, 0,
0.9586229266453453, 1.447539350583245, 2.147312047453889, 3.252815928229884, 3.505164024380132, 4.214706703453016, 4.883737290507347, 5.137625289439302, 6.026001920027076, 6.302444861804662, 7.056959246354084, 7.505592141977367, 7.768362503134963, 8.706680016658911, 9.117652890009725, 9.509967978515312, 10.05244414763490, 10.83734602766199, 11.10634047735888, 11.65985672070139, 11.99823917389867, 12.36665667490016, 12.94067224905971, 13.71250315329256, 14.01536861185948
