| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 4·11-s + 4·15-s + 4·17-s − 6·19-s + 4·23-s − 25-s − 4·27-s − 6·29-s + 4·31-s + 8·33-s + 6·37-s + 4·41-s − 12·43-s + 2·45-s + 12·47-s + 8·51-s + 6·53-s + 8·55-s − 12·57-s − 6·59-s + 6·61-s + 12·67-s + 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 1.03·15-s + 0.970·17-s − 1.37·19-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s + 1.39·33-s + 0.986·37-s + 0.624·41-s − 1.82·43-s + 0.298·45-s + 1.75·47-s + 1.12·51-s + 0.824·53-s + 1.07·55-s − 1.58·57-s − 0.781·59-s + 0.768·61-s + 1.46·67-s + 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.087045671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.087045671\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−12.86965821763452, −12.54268302719169, −11.88271491425718, −11.36799723951055, −11.02994519948566, −10.26384925630301, −9.931863594593997, −9.423533878499002, −9.173040642572038, −8.649219518917307, −8.238283643933574, −7.786408425642956, −7.131499570335037, −6.691392082171160, −6.193924504207357, −5.644041237484481, −5.292098333677617, −4.391464744140218, −3.960728548195974, −3.578391499142764, −2.823038350300093, −2.445385051153011, −1.840989555585654, −1.370135031306956, −0.6001144921227249,
0.6001144921227249, 1.370135031306956, 1.840989555585654, 2.445385051153011, 2.823038350300093, 3.578391499142764, 3.960728548195974, 4.391464744140218, 5.292098333677617, 5.644041237484481, 6.193924504207357, 6.691392082171160, 7.131499570335037, 7.786408425642956, 8.238283643933574, 8.649219518917307, 9.173040642572038, 9.423533878499002, 9.931863594593997, 10.26384925630301, 11.02994519948566, 11.36799723951055, 11.88271491425718, 12.54268302719169, 12.86965821763452
