| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 3·7-s + 8-s − 2·9-s + 12-s − 6·13-s + 3·14-s + 16-s − 7·17-s − 2·18-s − 5·19-s + 3·21-s + 6·23-s + 24-s − 6·26-s − 5·27-s + 3·28-s − 5·29-s − 3·31-s + 32-s − 7·34-s − 2·36-s − 3·37-s − 5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 1/4·16-s − 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.654·21-s + 1.25·23-s + 0.204·24-s − 1.17·26-s − 0.962·27-s + 0.566·28-s − 0.928·29-s − 0.538·31-s + 0.176·32-s − 1.20·34-s − 1/3·36-s − 0.493·37-s − 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55846710518097433809234529522, −7.15522824999053493291691011446, −6.25653525333154357749893379339, −5.35665492802824494420800719571, −4.74461890335682791263484236396, −4.24248350881714595932270404880, −3.13633289039528983743583004611, −2.32256149854148039372877464389, −1.86665970315543630585959813737, 0,
1.86665970315543630585959813737, 2.32256149854148039372877464389, 3.13633289039528983743583004611, 4.24248350881714595932270404880, 4.74461890335682791263484236396, 5.35665492802824494420800719571, 6.25653525333154357749893379339, 7.15522824999053493291691011446, 7.55846710518097433809234529522
