| L(s) = 1 | + 3·3-s − 5-s + 2·7-s + 6·9-s − 3·13-s − 3·15-s + 4·17-s − 4·19-s + 6·21-s − 23-s + 25-s + 9·27-s + 29-s + 31-s − 2·35-s − 8·37-s − 9·39-s + 11·41-s − 10·43-s − 6·45-s − 47-s − 3·49-s + 12·51-s − 6·53-s − 12·57-s − 8·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.755·7-s + 2·9-s − 0.832·13-s − 0.774·15-s + 0.970·17-s − 0.917·19-s + 1.30·21-s − 0.208·23-s + 1/5·25-s + 1.73·27-s + 0.185·29-s + 0.179·31-s − 0.338·35-s − 1.31·37-s − 1.44·39-s + 1.71·41-s − 1.52·43-s − 0.894·45-s − 0.145·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s − 1.58·57-s − 1.04·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.387014830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.387014830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.92086836632280153270529965910, −9.938539798352997483047810062433, −9.148863661306889990355175951583, −8.107227961593573582854075314832, −7.87730585194361309599526811936, −6.76826262654387773914920953864, −5.02078259089371613734136346031, −4.00989914232871951934465441985, −2.96124264604088800553202147952, −1.79937071104177429646005954852,
1.79937071104177429646005954852, 2.96124264604088800553202147952, 4.00989914232871951934465441985, 5.02078259089371613734136346031, 6.76826262654387773914920953864, 7.87730585194361309599526811936, 8.107227961593573582854075314832, 9.148863661306889990355175951583, 9.938539798352997483047810062433, 10.92086836632280153270529965910
