| L(s) = 1 | − 3-s + 2·5-s − 2·7-s − 2·9-s − 4·11-s + 4·13-s − 2·15-s + 7·17-s − 3·19-s + 2·21-s − 25-s + 5·27-s − 4·29-s + 6·31-s + 4·33-s − 4·35-s − 2·37-s − 4·39-s + 6·41-s − 5·43-s − 4·45-s − 10·47-s − 3·49-s − 7·51-s − 8·55-s + 3·57-s − 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s − 1.20·11-s + 1.10·13-s − 0.516·15-s + 1.69·17-s − 0.688·19-s + 0.436·21-s − 1/5·25-s + 0.962·27-s − 0.742·29-s + 1.07·31-s + 0.696·33-s − 0.676·35-s − 0.328·37-s − 0.640·39-s + 0.937·41-s − 0.762·43-s − 0.596·45-s − 1.45·47-s − 3/7·49-s − 0.980·51-s − 1.07·55-s + 0.397·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.338778953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.338778953\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.214001139306841520673590468067, −7.888843761882504412145231195891, −6.53848795677985654356751986853, −6.20759751467669929732999976969, −5.50258411308125337353905871045, −5.01418552993838718590970488054, −3.62083575014686785179152689460, −2.98263142873984303948509252085, −1.95994113893359951476236369619, −0.65998037406009917389791340426,
0.65998037406009917389791340426, 1.95994113893359951476236369619, 2.98263142873984303948509252085, 3.62083575014686785179152689460, 5.01418552993838718590970488054, 5.50258411308125337353905871045, 6.20759751467669929732999976969, 6.53848795677985654356751986853, 7.888843761882504412145231195891, 8.214001139306841520673590468067
