| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 11-s + 13-s + 15-s − 17-s + 3·19-s + 21-s − 6·23-s + 25-s + 5·27-s + 29-s + 7·31-s + 33-s + 35-s − 7·37-s − 39-s + 6·41-s + 4·43-s + 2·45-s − 6·47-s − 6·49-s + 51-s + 5·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.688·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.185·29-s + 1.25·31-s + 0.174·33-s + 0.169·35-s − 1.15·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 0.875·47-s − 6/7·49-s + 0.140·51-s + 0.686·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 17 T + p T^{2} \) | 1.89.ar |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.91440864506964, −14.26813707715826, −13.82308991019801, −13.48533564286422, −12.58757179654318, −12.25026930495475, −11.86668077345564, −11.20664924373369, −10.87874273058828, −10.25674095455098, −9.716506897778029, −9.126831999629770, −8.484664532425469, −7.988177425307290, −7.539959433844443, −6.680487895333409, −6.284824822797619, −5.779293777872292, −5.098542598525813, −4.581106792651636, −3.846408091880743, −3.165232044721456, −2.657873675094295, −1.713707241040880, −0.7405580217915563, 0,
0.7405580217915563, 1.713707241040880, 2.657873675094295, 3.165232044721456, 3.846408091880743, 4.581106792651636, 5.098542598525813, 5.779293777872292, 6.284824822797619, 6.680487895333409, 7.539959433844443, 7.988177425307290, 8.484664532425469, 9.126831999629770, 9.716506897778029, 10.25674095455098, 10.87874273058828, 11.20664924373369, 11.86668077345564, 12.25026930495475, 12.58757179654318, 13.48533564286422, 13.82308991019801, 14.26813707715826, 14.91440864506964
