L(s) = 1 | + 2·5-s − 2·7-s + 7·13-s − 17-s + 3·19-s − 6·23-s − 25-s − 2·29-s − 2·31-s − 4·35-s − 8·37-s + 2·41-s + 9·43-s − 7·47-s − 3·49-s + 10·53-s − 12·59-s + 12·61-s + 14·65-s − 5·67-s − 6·71-s + 6·73-s + 4·79-s + 7·83-s − 2·85-s + 3·89-s − 14·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.755·7-s + 1.94·13-s − 0.242·17-s + 0.688·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.676·35-s − 1.31·37-s + 0.312·41-s + 1.37·43-s − 1.02·47-s − 3/7·49-s + 1.37·53-s − 1.56·59-s + 1.53·61-s + 1.73·65-s − 0.610·67-s − 0.712·71-s + 0.702·73-s + 0.450·79-s + 0.768·83-s − 0.216·85-s + 0.317·89-s − 1.46·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.05728695053611, −12.52595148428033, −12.06430728593004, −11.50082175307439, −11.06608610586963, −10.60345412111397, −10.12080120237044, −9.767741663254332, −9.192904953214458, −8.932799826554965, −8.362558465796883, −7.868720572622599, −7.282396304205192, −6.734236620914425, −6.125191405379149, −6.021309369916319, −5.545906040160762, −4.926614813286769, −4.175929843660784, −3.552831113005120, −3.486442858042052, −2.591373066820552, −1.996132499099126, −1.516921356038569, −0.8485420089915492, 0,
0.8485420089915492, 1.516921356038569, 1.996132499099126, 2.591373066820552, 3.486442858042052, 3.552831113005120, 4.175929843660784, 4.926614813286769, 5.545906040160762, 6.021309369916319, 6.125191405379149, 6.734236620914425, 7.282396304205192, 7.868720572622599, 8.362558465796883, 8.932799826554965, 9.192904953214458, 9.767741663254332, 10.12080120237044, 10.60345412111397, 11.06608610586963, 11.50082175307439, 12.06430728593004, 12.52595148428033, 13.05728695053611