| L(s) = 1 | − 5-s + 7-s + 3·11-s − 6·13-s − 2·23-s + 25-s + 6·29-s − 35-s − 37-s + 9·41-s − 10·43-s − 47-s − 6·49-s − 53-s − 3·55-s − 12·61-s + 6·65-s + 5·71-s + 3·73-s + 3·77-s + 16·79-s − 11·83-s − 6·91-s + 8·97-s + 101-s − 10·103-s − 20·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.904·11-s − 1.66·13-s − 0.417·23-s + 1/5·25-s + 1.11·29-s − 0.169·35-s − 0.164·37-s + 1.40·41-s − 1.52·43-s − 0.145·47-s − 6/7·49-s − 0.137·53-s − 0.404·55-s − 1.53·61-s + 0.744·65-s + 0.593·71-s + 0.351·73-s + 0.341·77-s + 1.80·79-s − 1.20·83-s − 0.628·91-s + 0.812·97-s + 0.0995·101-s − 0.985·103-s − 1.93·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−7.79638999644460481713320026732, −6.87430274668279320068239669505, −6.44738044605106477199714290862, −5.36200728272846905543618876830, −4.71505583812144967193498673721, −4.12479975688139803033309734468, −3.16162875287926518278369267338, −2.31757562612808024695711287089, −1.29315237816564448936786559182, 0,
1.29315237816564448936786559182, 2.31757562612808024695711287089, 3.16162875287926518278369267338, 4.12479975688139803033309734468, 4.71505583812144967193498673721, 5.36200728272846905543618876830, 6.44738044605106477199714290862, 6.87430274668279320068239669505, 7.79638999644460481713320026732
