| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 11-s + 12-s − 13-s + 14-s + 16-s − 5·17-s − 2·18-s + 3·19-s + 21-s + 22-s − 4·23-s + 24-s − 26-s − 5·27-s + 28-s − 9·29-s − 11·31-s + 32-s + 33-s − 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.471·18-s + 0.688·19-s + 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.962·27-s + 0.188·28-s − 1.67·29-s − 1.97·31-s + 0.176·32-s + 0.174·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.46316845749760788662508415023, −7.01506446981039755747597921766, −5.93623229399867005222989276210, −5.54043486522975993955726719518, −4.67447011824506761277198307151, −3.84206275454317157079632472326, −3.31582867841607494463298023158, −2.26307532894800228691571498663, −1.76164583064395163180285016449, 0,
1.76164583064395163180285016449, 2.26307532894800228691571498663, 3.31582867841607494463298023158, 3.84206275454317157079632472326, 4.67447011824506761277198307151, 5.54043486522975993955726719518, 5.93623229399867005222989276210, 7.01506446981039755747597921766, 7.46316845749760788662508415023
