| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s + 10-s − 11-s + 4·14-s + 16-s − 6·17-s − 4·19-s − 20-s + 22-s + 4·23-s + 25-s − 4·28-s + 2·29-s − 8·31-s − 32-s + 6·34-s + 4·35-s − 2·37-s + 4·38-s + 40-s + 6·41-s − 4·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.06·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s − 0.755·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.676·35-s − 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-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 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−13.54252391534842, −13.04361319770487, −12.80604206560321, −12.38345967588334, −11.73254130680207, −11.22614870932043, −10.74378791048194, −10.43915321460795, −9.899835784828027, −9.294096395919230, −8.835671215192523, −8.742580440558628, −7.954347983505373, −7.299705980755430, −6.925692073645408, −6.651537683133136, −5.933123267897951, −5.581273133661887, −4.695467058900806, −4.134102332760740, −3.664904363145391, −2.907044626596108, −2.581712095377446, −1.888783370272288, −0.9900551203252667, 0, 0,
0.9900551203252667, 1.888783370272288, 2.581712095377446, 2.907044626596108, 3.664904363145391, 4.134102332760740, 4.695467058900806, 5.581273133661887, 5.933123267897951, 6.651537683133136, 6.925692073645408, 7.299705980755430, 7.954347983505373, 8.742580440558628, 8.835671215192523, 9.294096395919230, 9.899835784828027, 10.43915321460795, 10.74378791048194, 11.22614870932043, 11.73254130680207, 12.38345967588334, 12.80604206560321, 13.04361319770487, 13.54252391534842
