| L(s) = 1 | − 2·3-s − 2·5-s − 7-s + 9-s − 2·11-s + 4·13-s + 4·15-s + 2·21-s − 23-s − 25-s + 4·27-s + 2·29-s + 4·33-s + 2·35-s + 4·37-s − 8·39-s − 6·41-s + 2·43-s − 2·45-s + 4·47-s + 49-s + 4·55-s + 2·59-s + 10·61-s − 63-s − 8·65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 1.03·15-s + 0.436·21-s − 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.696·33-s + 0.338·35-s + 0.657·37-s − 1.28·39-s − 0.937·41-s + 0.304·43-s − 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.539·55-s + 0.260·59-s + 1.28·61-s − 0.125·63-s − 0.992·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9949513184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9949513184\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 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
−12.42130465114355, −11.95225663823754, −11.47133268489392, −11.31937422506025, −10.76526765235114, −10.38707654847248, −9.958076801384824, −9.374604981467367, −8.716885114942959, −8.395509923670111, −7.890511864248889, −7.463177756411114, −6.828863493931991, −6.394779080284705, −6.088195889178337, −5.414228006851993, −5.163264037016197, −4.532212446817327, −3.800627225329622, −3.753201399960588, −2.892777378108562, −2.398757330432996, −1.529832152606292, −0.7863591500315223, −0.3924161213512054,
0.3924161213512054, 0.7863591500315223, 1.529832152606292, 2.398757330432996, 2.892777378108562, 3.753201399960588, 3.800627225329622, 4.532212446817327, 5.163264037016197, 5.414228006851993, 6.088195889178337, 6.394779080284705, 6.828863493931991, 7.463177756411114, 7.890511864248889, 8.395509923670111, 8.716885114942959, 9.374604981467367, 9.958076801384824, 10.38707654847248, 10.76526765235114, 11.31937422506025, 11.47133268489392, 11.95225663823754, 12.42130465114355
