| L(s) = 1 | + 5-s + 4·7-s + 2·11-s − 13-s − 2·17-s + 6·19-s − 6·23-s + 25-s − 2·29-s + 10·31-s + 4·35-s − 2·37-s + 6·41-s − 10·43-s + 4·47-s + 9·49-s − 2·53-s + 2·55-s + 6·59-s + 2·61-s − 65-s + 4·67-s + 6·71-s − 6·73-s + 8·77-s + 12·79-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.603·11-s − 0.277·13-s − 0.485·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s − 1.52·43-s + 0.583·47-s + 9/7·49-s − 0.274·53-s + 0.269·55-s + 0.781·59-s + 0.256·61-s − 0.124·65-s + 0.488·67-s + 0.712·71-s − 0.702·73-s + 0.911·77-s + 1.35·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.107791900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.107791900\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.74626382643602401855892436666, −7.10424803507470864464940101308, −6.29445865007141718674511496603, −5.60494696841649733936617387816, −4.90107081547904170536801789220, −4.38402762445566525105146722688, −3.48550600608682849609112272424, −2.42000959348824453784320811929, −1.73660368193522923800787188800, −0.898271805730672484896419980636,
0.898271805730672484896419980636, 1.73660368193522923800787188800, 2.42000959348824453784320811929, 3.48550600608682849609112272424, 4.38402762445566525105146722688, 4.90107081547904170536801789220, 5.60494696841649733936617387816, 6.29445865007141718674511496603, 7.10424803507470864464940101308, 7.74626382643602401855892436666
