| L(s) = 1 | − 5-s + 3·11-s + 13-s − 6·17-s − 8·19-s + 6·23-s − 4·25-s − 5·29-s + 9·31-s + 8·37-s + 8·41-s − 8·43-s + 12·47-s − 5·53-s − 3·55-s + 15·59-s − 65-s − 4·67-s + 14·71-s − 2·73-s + 11·79-s − 13·83-s + 6·85-s − 4·89-s + 8·95-s + 5·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.904·11-s + 0.277·13-s − 1.45·17-s − 1.83·19-s + 1.25·23-s − 4/5·25-s − 0.928·29-s + 1.61·31-s + 1.31·37-s + 1.24·41-s − 1.21·43-s + 1.75·47-s − 0.686·53-s − 0.404·55-s + 1.95·59-s − 0.124·65-s − 0.488·67-s + 1.66·71-s − 0.234·73-s + 1.23·79-s − 1.42·83-s + 0.650·85-s − 0.423·89-s + 0.820·95-s + 0.507·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396786018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396786018\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−12.57268432393766, −12.01064484912418, −11.42954717086259, −11.25198946747411, −10.84194954583758, −10.35133431505957, −9.631666208322791, −9.373867944237390, −8.771462672358245, −8.443333351614799, −8.103552475116769, −7.329025278884681, −6.979045717113692, −6.447875285091482, −6.156285443627228, −5.615617398268272, −4.783486605450031, −4.382118167400332, −4.084463664929782, −3.595878766050675, −2.802757737264580, −2.276676436353988, −1.851706876490083, −0.9353311918974349, −0.4740385656947347,
0.4740385656947347, 0.9353311918974349, 1.851706876490083, 2.276676436353988, 2.802757737264580, 3.595878766050675, 4.084463664929782, 4.382118167400332, 4.783486605450031, 5.615617398268272, 6.156285443627228, 6.447875285091482, 6.979045717113692, 7.329025278884681, 8.103552475116769, 8.443333351614799, 8.771462672358245, 9.373867944237390, 9.631666208322791, 10.35133431505957, 10.84194954583758, 11.25198946747411, 11.42954717086259, 12.01064484912418, 12.57268432393766
