| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s − 7·13-s + 16-s + 4·17-s − 18-s + 19-s + 22-s − 23-s + 24-s + 7·26-s − 27-s − 8·29-s + 6·31-s − 32-s + 33-s − 4·34-s + 36-s + 3·37-s − 38-s + 7·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.94·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 1.37·26-s − 0.192·27-s − 1.48·29-s + 1.07·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + 1/6·36-s + 0.493·37-s − 0.162·38-s + 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−7.49082035325798808627589780843, −7.17859849879754654403995664249, −6.12743460060459553474267724678, −5.55929255512570164244895593900, −4.85906600486342774120768515337, −4.01264780578647336422952714188, −2.86462309925535509293443510036, −2.21471342708607028514697493587, −1.03446535266972550117746122432, 0,
1.03446535266972550117746122432, 2.21471342708607028514697493587, 2.86462309925535509293443510036, 4.01264780578647336422952714188, 4.85906600486342774120768515337, 5.55929255512570164244895593900, 6.12743460060459553474267724678, 7.17859849879754654403995664249, 7.49082035325798808627589780843
