| L(s) = 1 | − 4·11-s − 6·13-s − 17-s − 4·19-s + 8·23-s + 6·29-s − 4·31-s − 2·37-s − 6·41-s − 4·43-s − 4·47-s − 7·49-s + 10·53-s + 12·59-s + 6·61-s − 4·67-s − 14·73-s + 4·79-s − 16·83-s − 10·89-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 1.66·13-s − 0.242·17-s − 0.917·19-s + 1.66·23-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.583·47-s − 49-s + 1.37·53-s + 1.56·59-s + 0.768·61-s − 0.488·67-s − 1.63·73-s + 0.450·79-s − 1.75·83-s − 1.05·89-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6018188675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6018188675\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.55628056171763, −13.62322942967545, −13.25652245825669, −12.73023308675192, −12.47607934796145, −11.65044057506414, −11.37045987546278, −10.53288214867161, −10.28305013295474, −9.824077096655359, −9.131929474445044, −8.515380601398627, −8.194374648277688, −7.430457567752277, −6.910841060460756, −6.700943760652488, −5.602959890233755, −5.228641845522092, −4.766218398432560, −4.197376049393322, −3.243640403348241, −2.697724358046936, −2.262790612112185, −1.368776810760981, −0.2574766026913279,
0.2574766026913279, 1.368776810760981, 2.262790612112185, 2.697724358046936, 3.243640403348241, 4.197376049393322, 4.766218398432560, 5.228641845522092, 5.602959890233755, 6.700943760652488, 6.910841060460756, 7.430457567752277, 8.194374648277688, 8.515380601398627, 9.131929474445044, 9.824077096655359, 10.28305013295474, 10.53288214867161, 11.37045987546278, 11.65044057506414, 12.47607934796145, 12.73023308675192, 13.25652245825669, 13.62322942967545, 14.55628056171763
