| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 5·7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s + 5·14-s − 15-s + 16-s − 8·17-s − 18-s − 5·19-s − 20-s − 5·21-s + 3·22-s − 4·23-s − 24-s + 25-s + 27-s − 5·28-s − 4·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.223·20-s − 1.09·21-s + 0.639·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.944·28-s − 0.742·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3157229217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3157229217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−8.389753298913443470853712944656, −7.55164127065664012103482968139, −6.82677550398519109098126487887, −6.43171013588683796970282036283, −5.50131141718180918874804530132, −4.19575733074684163397955771306, −3.66840913133030569049909060586, −2.62344742699262395237953220949, −2.17170518156627208725673846946, −0.30097374533056560328987907482,
0.30097374533056560328987907482, 2.17170518156627208725673846946, 2.62344742699262395237953220949, 3.66840913133030569049909060586, 4.19575733074684163397955771306, 5.50131141718180918874804530132, 6.43171013588683796970282036283, 6.82677550398519109098126487887, 7.55164127065664012103482968139, 8.389753298913443470853712944656
