| L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s − 2·13-s + 4·15-s + 4·17-s + 3·19-s − 21-s − 2·23-s + 11·25-s + 27-s + 6·29-s + 5·31-s − 4·35-s + 3·37-s − 2·39-s − 2·41-s − 12·43-s + 4·45-s − 2·47-s − 6·49-s + 4·51-s + 6·53-s + 3·57-s + 10·59-s + 3·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.688·19-s − 0.218·21-s − 0.417·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 0.898·31-s − 0.676·35-s + 0.493·37-s − 0.320·39-s − 0.312·41-s − 1.82·43-s + 0.596·45-s − 0.291·47-s − 6/7·49-s + 0.560·51-s + 0.824·53-s + 0.397·57-s + 1.30·59-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.716297501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.716297501\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−8.270084621494176042041252662882, −7.32057142730538592585513756279, −6.62894911055050850603554095397, −6.00175857633146874298631346404, −5.28227304204569921820826185035, −4.63110352385300812194491410922, −3.34963292037783635548971760769, −2.76335125573149397615768643500, −1.94820519885700493081501567293, −1.05469196837130809633560747721,
1.05469196837130809633560747721, 1.94820519885700493081501567293, 2.76335125573149397615768643500, 3.34963292037783635548971760769, 4.63110352385300812194491410922, 5.28227304204569921820826185035, 6.00175857633146874298631346404, 6.62894911055050850603554095397, 7.32057142730538592585513756279, 8.270084621494176042041252662882
