| L(s) = 1 | + 2·5-s − 2·7-s − 4·11-s − 2·13-s + 2·17-s + 2·19-s + 23-s − 25-s + 4·29-s − 4·35-s + 2·37-s + 2·43-s − 12·47-s − 3·49-s − 2·53-s − 8·55-s − 12·59-s − 14·61-s − 4·65-s − 2·67-s + 6·73-s + 8·77-s − 6·79-s − 4·83-s + 4·85-s − 18·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.458·19-s + 0.208·23-s − 1/5·25-s + 0.742·29-s − 0.676·35-s + 0.328·37-s + 0.304·43-s − 1.75·47-s − 3/7·49-s − 0.274·53-s − 1.07·55-s − 1.56·59-s − 1.79·61-s − 0.496·65-s − 0.244·67-s + 0.702·73-s + 0.911·77-s − 0.675·79-s − 0.439·83-s + 0.433·85-s − 1.90·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−8.147333055433711887481828188549, −7.58527073186463995761840691517, −6.66319955417652553644737431184, −5.97675172072583571635092589554, −5.29178061599942647616704041474, −4.55265856611633239781554096866, −3.19849046847586695687233771210, −2.69168994626663399567774322805, −1.55196167858596033257134738316, 0,
1.55196167858596033257134738316, 2.69168994626663399567774322805, 3.19849046847586695687233771210, 4.55265856611633239781554096866, 5.29178061599942647616704041474, 5.97675172072583571635092589554, 6.66319955417652553644737431184, 7.58527073186463995761840691517, 8.147333055433711887481828188549
