| L(s) = 1 | + 3·3-s + 2·5-s − 7-s + 6·9-s + 2·11-s + 6·15-s + 19-s − 3·21-s + 23-s − 25-s + 9·27-s − 29-s − 3·31-s + 6·33-s − 2·35-s − 2·37-s + 6·41-s + 2·43-s + 12·45-s − 47-s + 49-s − 7·53-s + 4·55-s + 3·57-s + 5·59-s − 12·61-s − 6·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s − 0.377·7-s + 2·9-s + 0.603·11-s + 1.54·15-s + 0.229·19-s − 0.654·21-s + 0.208·23-s − 1/5·25-s + 1.73·27-s − 0.185·29-s − 0.538·31-s + 1.04·33-s − 0.338·35-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 1.78·45-s − 0.145·47-s + 1/7·49-s − 0.961·53-s + 0.539·55-s + 0.397·57-s + 0.650·59-s − 1.53·61-s − 0.755·63-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−12.99696155805139, −12.34954912691300, −12.01053803586098, −11.34576659834962, −10.66898199646093, −10.39583795869586, −9.734061487485344, −9.372815923948667, −9.281884571292421, −8.699660077514387, −8.291326144462187, −7.652626506566513, −7.340670242264791, −6.884465081588990, −6.181258553532346, −5.921807713150558, −5.240610743761671, −4.525768154748569, −4.129575773159313, −3.558073586512651, −3.063947949281236, −2.681467106785270, −2.042227933107593, −1.605159183416547, −1.118133823616930, 0,
1.118133823616930, 1.605159183416547, 2.042227933107593, 2.681467106785270, 3.063947949281236, 3.558073586512651, 4.129575773159313, 4.525768154748569, 5.240610743761671, 5.921807713150558, 6.181258553532346, 6.884465081588990, 7.340670242264791, 7.652626506566513, 8.291326144462187, 8.699660077514387, 9.281884571292421, 9.372815923948667, 9.734061487485344, 10.39583795869586, 10.66898199646093, 11.34576659834962, 12.01053803586098, 12.34954912691300, 12.99696155805139
