| L(s) = 1 | + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 11-s + 2·14-s + 16-s + 2·17-s − 20-s + 22-s − 6·23-s + 25-s + 2·28-s − 4·29-s + 4·31-s + 32-s + 2·34-s − 2·35-s + 8·37-s − 40-s + 6·41-s + 4·43-s + 44-s − 6·46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.377·28-s − 0.742·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.338·35-s + 1.31·37-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.150·44-s − 0.884·46-s − 1.75·47-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 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.41396823584421, −13.01545727627191, −12.49941314517534, −12.05903836823127, −11.59311536539934, −11.23364797394729, −10.87754343995991, −10.17073719675789, −9.694198251258992, −9.286481026897457, −8.364173036191843, −8.205676317789139, −7.575745461265031, −7.318127946596939, −6.481715594064450, −6.058152738939432, −5.672633975779798, −4.853485029912630, −4.589343776039363, −4.014607229587242, −3.521948470881748, −2.878444777686966, −2.244958188993287, −1.594480335700127, −0.9975785724240999, 0,
0.9975785724240999, 1.594480335700127, 2.244958188993287, 2.878444777686966, 3.521948470881748, 4.014607229587242, 4.589343776039363, 4.853485029912630, 5.672633975779798, 6.058152738939432, 6.481715594064450, 7.318127946596939, 7.575745461265031, 8.205676317789139, 8.364173036191843, 9.286481026897457, 9.694198251258992, 10.17073719675789, 10.87754343995991, 11.23364797394729, 11.59311536539934, 12.05903836823127, 12.49941314517534, 13.01545727627191, 13.41396823584421
