| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 2·7-s − 8-s + 9-s + 2·12-s − 6·13-s + 2·14-s + 16-s + 17-s − 18-s + 8·19-s − 4·21-s + 2·23-s − 2·24-s + 6·26-s − 4·27-s − 2·28-s − 6·29-s − 2·31-s − 32-s − 34-s + 36-s − 6·37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.66·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.872·21-s + 0.417·23-s − 0.408·24-s + 1.17·26-s − 0.769·27-s − 0.377·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.986·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.03928967427852, −13.49190150328114, −13.10154805749295, −12.48396415577272, −11.93608989597843, −11.65860180195686, −10.92499985950922, −10.29649653809974, −9.708754195534198, −9.530177320909158, −9.225130785342782, −8.517774526873698, −7.995888144044222, −7.561237132851788, −7.055133423691255, −6.798480208158294, −5.821938217079988, −5.273980224404798, −4.882316398342087, −3.655432892656869, −3.499341280017172, −2.886437298833928, −2.291216318165431, −1.799157822399621, −0.8047622861649161, 0,
0.8047622861649161, 1.799157822399621, 2.291216318165431, 2.886437298833928, 3.499341280017172, 3.655432892656869, 4.882316398342087, 5.273980224404798, 5.821938217079988, 6.798480208158294, 7.055133423691255, 7.561237132851788, 7.995888144044222, 8.517774526873698, 9.225130785342782, 9.530177320909158, 9.708754195534198, 10.29649653809974, 10.92499985950922, 11.65860180195686, 11.93608989597843, 12.48396415577272, 13.10154805749295, 13.49190150328114, 14.03928967427852
