| L(s) = 1 | + 5-s + 3·7-s − 3·11-s − 2·17-s − 6·19-s + 3·23-s − 4·25-s − 4·29-s − 6·31-s + 3·35-s − 7·37-s + 5·41-s − 6·43-s − 9·47-s + 2·49-s + 11·53-s − 3·55-s − 6·59-s − 8·61-s + 12·71-s − 15·73-s − 9·77-s + 12·79-s − 3·83-s − 2·85-s + 2·89-s − 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.904·11-s − 0.485·17-s − 1.37·19-s + 0.625·23-s − 4/5·25-s − 0.742·29-s − 1.07·31-s + 0.507·35-s − 1.15·37-s + 0.780·41-s − 0.914·43-s − 1.31·47-s + 2/7·49-s + 1.51·53-s − 0.404·55-s − 0.781·59-s − 1.02·61-s + 1.42·71-s − 1.75·73-s − 1.02·77-s + 1.35·79-s − 0.329·83-s − 0.216·85-s + 0.211·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7846702062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7846702062\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.99637940901007, −12.60622139251120, −11.93413550322560, −11.48809562742697, −10.98620640533763, −10.66486732844851, −10.31896895822918, −9.635607355982286, −9.148647205025261, −8.650991611943934, −8.286413422672887, −7.691015407957031, −7.398158129234249, −6.683448120158350, −6.248478222912570, −5.610598499860759, −5.120773136094473, −4.874033386751301, −4.115149428667119, −3.709876541607460, −2.883731823525287, −2.273226483493495, −1.827749126988435, −1.398483259919609, −0.2261880626493817,
0.2261880626493817, 1.398483259919609, 1.827749126988435, 2.273226483493495, 2.883731823525287, 3.709876541607460, 4.115149428667119, 4.874033386751301, 5.120773136094473, 5.610598499860759, 6.248478222912570, 6.683448120158350, 7.398158129234249, 7.691015407957031, 8.286413422672887, 8.650991611943934, 9.148647205025261, 9.635607355982286, 10.31896895822918, 10.66486732844851, 10.98620640533763, 11.48809562742697, 11.93413550322560, 12.60622139251120, 12.99637940901007
