| L(s) = 1 | − 3-s + 9-s − 11-s − 7·13-s − 17-s − 3·19-s + 9·23-s − 27-s + 6·29-s − 4·31-s + 33-s + 10·37-s + 7·39-s − 3·41-s − 3·43-s − 8·47-s + 51-s + 4·53-s + 3·57-s − 12·59-s + 6·61-s + 8·67-s − 9·69-s − 8·71-s − 10·73-s + 81-s + 10·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.94·13-s − 0.242·17-s − 0.688·19-s + 1.87·23-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.174·33-s + 1.64·37-s + 1.12·39-s − 0.468·41-s − 0.457·43-s − 1.16·47-s + 0.140·51-s + 0.549·53-s + 0.397·57-s − 1.56·59-s + 0.768·61-s + 0.977·67-s − 1.08·69-s − 0.949·71-s − 1.17·73-s + 1/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.09445958468609, −12.65735644837388, −12.05528800895327, −11.83796855785842, −11.15684340142370, −10.85195610551551, −10.31725199201190, −9.889937053152765, −9.430456451081164, −8.995411541558745, −8.382229472835370, −7.810538351480695, −7.414961271552457, −6.800011531759316, −6.632468953363002, −5.907814017321212, −5.295951911529886, −4.824291370515906, −4.658672385905390, −3.966359808931184, −3.082962955608933, −2.733783345737631, −2.152659428359070, −1.418726891341128, −0.6419549884822385, 0,
0.6419549884822385, 1.418726891341128, 2.152659428359070, 2.733783345737631, 3.082962955608933, 3.966359808931184, 4.658672385905390, 4.824291370515906, 5.295951911529886, 5.907814017321212, 6.632468953363002, 6.800011531759316, 7.414961271552457, 7.810538351480695, 8.382229472835370, 8.995411541558745, 9.430456451081164, 9.889937053152765, 10.31725199201190, 10.85195610551551, 11.15684340142370, 11.83796855785842, 12.05528800895327, 12.65735644837388, 13.09445958468609
