| L(s) = 1 | + 2-s + 4-s − 5-s + 5·7-s + 8-s − 10-s − 11-s + 5·14-s + 16-s − 6·17-s + 2·19-s − 20-s − 22-s − 3·23-s + 25-s + 5·28-s + 3·29-s + 5·31-s + 32-s − 6·34-s − 5·35-s + 2·37-s + 2·38-s − 40-s − 3·41-s − 43-s − 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.88·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.33·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s + 0.944·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 1.02·34-s − 0.845·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s − 0.468·41-s − 0.152·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.56127744826883, −13.13631431256339, −12.32655564357687, −12.11089373619842, −11.47982874355415, −11.30426813112607, −10.82785212353626, −10.32761207966642, −9.811273517088007, −8.892017725499240, −8.647108883814629, −8.039317878745261, −7.728503776723715, −7.224210937587114, −6.595253220394556, −6.108344823760156, −5.409691196495726, −4.918112222672766, −4.592824859095475, −4.140646924294471, −3.573293806818364, −2.618531190423615, −2.369038921604365, −1.579386168925663, −1.059063650791291, 0,
1.059063650791291, 1.579386168925663, 2.369038921604365, 2.618531190423615, 3.573293806818364, 4.140646924294471, 4.592824859095475, 4.918112222672766, 5.409691196495726, 6.108344823760156, 6.595253220394556, 7.224210937587114, 7.728503776723715, 8.039317878745261, 8.647108883814629, 8.892017725499240, 9.811273517088007, 10.32761207966642, 10.82785212353626, 11.30426813112607, 11.47982874355415, 12.11089373619842, 12.32655564357687, 13.13631431256339, 13.56127744826883
