| L(s) = 1 | − 5-s − 3·7-s − 5·11-s − 13-s − 3·17-s + 8·19-s + 3·23-s + 25-s − 2·29-s + 10·31-s + 3·35-s − 3·37-s + 3·41-s + 10·43-s + 6·47-s + 2·49-s + 5·53-s + 5·55-s − 4·59-s − 7·61-s + 65-s − 12·67-s − 5·71-s + 2·73-s + 15·77-s + 13·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 1.50·11-s − 0.277·13-s − 0.727·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s − 0.371·29-s + 1.79·31-s + 0.507·35-s − 0.493·37-s + 0.468·41-s + 1.52·43-s + 0.875·47-s + 2/7·49-s + 0.686·53-s + 0.674·55-s − 0.520·59-s − 0.896·61-s + 0.124·65-s − 1.46·67-s − 0.593·71-s + 0.234·73-s + 1.70·77-s + 1.46·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 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.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.62295398423809722416148191494, −6.73318962792036887738672739379, −6.00920692302895758664849327025, −5.27281214555356199649075773159, −4.65977087241876803139635842891, −3.73190666639864770142781372908, −2.87194243113059632150745464072, −2.58267220446043774760283782667, −0.994673613006434617934448839279, 0,
0.994673613006434617934448839279, 2.58267220446043774760283782667, 2.87194243113059632150745464072, 3.73190666639864770142781372908, 4.65977087241876803139635842891, 5.27281214555356199649075773159, 6.00920692302895758664849327025, 6.73318962792036887738672739379, 7.62295398423809722416148191494
