| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s + 11-s + 2·13-s + 15-s + 6·17-s − 4·19-s + 4·21-s + 8·23-s + 25-s + 27-s − 10·29-s + 33-s + 4·35-s − 10·37-s + 2·39-s − 6·41-s + 8·43-s + 45-s + 9·49-s + 6·51-s − 2·53-s + 55-s − 4·57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.174·33-s + 0.676·35-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.244774228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.244774228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−8.906295971457828905632973564068, −8.098228457124124193368174070659, −7.53056689548850983337487483941, −6.67878192703296195936770080674, −5.56839229863263806855724175036, −5.03983804702962428085166075223, −4.02177543299586824926644792552, −3.16402664617424669291998192709, −1.92405669509802814559295295410, −1.28766279884774117892017161952,
1.28766279884774117892017161952, 1.92405669509802814559295295410, 3.16402664617424669291998192709, 4.02177543299586824926644792552, 5.03983804702962428085166075223, 5.56839229863263806855724175036, 6.67878192703296195936770080674, 7.53056689548850983337487483941, 8.098228457124124193368174070659, 8.906295971457828905632973564068
