| L(s) = 1 | + 2·2-s + 2·4-s + 7-s + 3·11-s + 2·14-s − 4·16-s + 6·17-s + 6·19-s + 6·22-s + 2·23-s + 2·28-s − 10·29-s − 4·31-s − 8·32-s + 12·34-s − 2·37-s + 12·38-s − 8·41-s + 2·43-s + 6·44-s + 4·46-s + 49-s − 14·53-s − 20·58-s + 6·59-s + 10·61-s − 8·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s + 0.904·11-s + 0.534·14-s − 16-s + 1.45·17-s + 1.37·19-s + 1.27·22-s + 0.417·23-s + 0.377·28-s − 1.85·29-s − 0.718·31-s − 1.41·32-s + 2.05·34-s − 0.328·37-s + 1.94·38-s − 1.24·41-s + 0.304·43-s + 0.904·44-s + 0.589·46-s + 1/7·49-s − 1.92·53-s − 2.62·58-s + 0.781·59-s + 1.28·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.89987626693609, −12.62373306520362, −12.29903037893235, −11.55781134757218, −11.43752449745519, −11.13696249429660, −10.27437672927809, −9.756237350729068, −9.358813192925073, −8.982095704940716, −8.226074026131121, −7.782560499981618, −7.221018909951024, −6.811229782901302, −6.319733000674849, −5.557426370991700, −5.303168868404144, −5.147813453387226, −4.231035481469760, −3.766377655044552, −3.462127825601112, −2.987805002388780, −2.199473269999729, −1.553304556098053, −1.044790174505003, 0,
1.044790174505003, 1.553304556098053, 2.199473269999729, 2.987805002388780, 3.462127825601112, 3.766377655044552, 4.231035481469760, 5.147813453387226, 5.303168868404144, 5.557426370991700, 6.319733000674849, 6.811229782901302, 7.221018909951024, 7.782560499981618, 8.226074026131121, 8.982095704940716, 9.358813192925073, 9.756237350729068, 10.27437672927809, 11.13696249429660, 11.43752449745519, 11.55781134757218, 12.29903037893235, 12.62373306520362, 12.89987626693609
