| L(s) = 1 | − 2-s − 4-s + 3·8-s − 5·11-s − 4·13-s − 16-s − 4·17-s − 19-s + 5·22-s − 9·23-s + 4·26-s − 7·29-s + 3·31-s − 5·32-s + 4·34-s + 10·37-s + 38-s + 2·41-s − 4·43-s + 5·44-s + 9·46-s + 8·47-s − 7·49-s + 4·52-s + 11·53-s + 7·58-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.50·11-s − 1.10·13-s − 1/4·16-s − 0.970·17-s − 0.229·19-s + 1.06·22-s − 1.87·23-s + 0.784·26-s − 1.29·29-s + 0.538·31-s − 0.883·32-s + 0.685·34-s + 1.64·37-s + 0.162·38-s + 0.312·41-s − 0.609·43-s + 0.753·44-s + 1.32·46-s + 1.16·47-s − 49-s + 0.554·52-s + 1.51·53-s + 0.919·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3688898201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3688898201\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.376378245347935148437507042550, −7.68482790525340168661953259913, −7.37277090191370257948528222050, −6.16222303992150708987153499251, −5.38684477330525658190410469346, −4.60751892259704265206768938494, −4.00808740747156889119723280689, −2.63019889004286863265898711797, −1.96632877507441966448678532296, −0.36381805109857369562985824118,
0.36381805109857369562985824118, 1.96632877507441966448678532296, 2.63019889004286863265898711797, 4.00808740747156889119723280689, 4.60751892259704265206768938494, 5.38684477330525658190410469346, 6.16222303992150708987153499251, 7.37277090191370257948528222050, 7.68482790525340168661953259913, 8.376378245347935148437507042550
