| L(s) = 1 | − 5-s − 3·7-s − 5·11-s − 4·13-s + 8·17-s + 2·19-s + 2·23-s − 4·25-s + 6·29-s + 7·31-s + 3·35-s + 6·37-s + 6·41-s − 2·43-s + 6·47-s + 2·49-s + 5·53-s + 5·55-s + 4·59-s + 8·61-s + 4·65-s − 10·67-s − 8·71-s + 73-s + 15·77-s − 16·79-s + 11·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.11·29-s + 1.25·31-s + 0.507·35-s + 0.986·37-s + 0.937·41-s − 0.304·43-s + 0.875·47-s + 2/7·49-s + 0.686·53-s + 0.674·55-s + 0.520·59-s + 1.02·61-s + 0.496·65-s − 1.22·67-s − 0.949·71-s + 0.117·73-s + 1.70·77-s − 1.80·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.069689778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.069689778\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.649745872944478294591396163713, −8.352104784585146468182630520222, −7.67996846591970940803333630612, −7.15340015105356378963058851080, −5.98276615954030285758204092167, −5.32932694357691635729819844111, −4.34190910531857724149474492775, −3.11556422792398663255142778514, −2.66387606752889082799803334542, −0.68688623656136131006877383651,
0.68688623656136131006877383651, 2.66387606752889082799803334542, 3.11556422792398663255142778514, 4.34190910531857724149474492775, 5.32932694357691635729819844111, 5.98276615954030285758204092167, 7.15340015105356378963058851080, 7.67996846591970940803333630612, 8.352104784585146468182630520222, 9.649745872944478294591396163713
