| L(s) = 1 | + 5-s + 2·7-s + 3·11-s + 5·13-s + 3·17-s + 7·23-s + 25-s − 5·29-s − 5·31-s + 2·35-s + 6·37-s + 9·43-s + 9·47-s − 3·49-s + 8·53-s + 3·55-s − 4·59-s − 14·61-s + 5·65-s + 12·67-s − 8·71-s − 2·73-s + 6·77-s − 11·79-s + 6·83-s + 3·85-s + 10·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.904·11-s + 1.38·13-s + 0.727·17-s + 1.45·23-s + 1/5·25-s − 0.928·29-s − 0.898·31-s + 0.338·35-s + 0.986·37-s + 1.37·43-s + 1.31·47-s − 3/7·49-s + 1.09·53-s + 0.404·55-s − 0.520·59-s − 1.79·61-s + 0.620·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s + 0.683·77-s − 1.23·79-s + 0.658·83-s + 0.325·85-s + 1.04·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.333463841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333463841\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 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 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.66140808153721591025214706319, −7.19974071948624865643231230692, −6.23821848458239971739185088390, −5.79179688832438790459366172329, −5.07128659575528921484081003283, −4.16617067668056656069671336060, −3.58389131704070409692076730155, −2.62148986519053440586599240678, −1.52379602117965217849632647733, −1.03014242269710431224333446559,
1.03014242269710431224333446559, 1.52379602117965217849632647733, 2.62148986519053440586599240678, 3.58389131704070409692076730155, 4.16617067668056656069671336060, 5.07128659575528921484081003283, 5.79179688832438790459366172329, 6.23821848458239971739185088390, 7.19974071948624865643231230692, 7.66140808153721591025214706319
