| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 2·11-s + 2·13-s + 14-s + 16-s + 6·19-s + 20-s + 2·22-s − 6·23-s + 25-s + 2·26-s + 28-s + 2·29-s + 32-s + 35-s − 12·37-s + 6·38-s + 40-s − 10·41-s − 8·43-s + 2·44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s + 0.176·32-s + 0.169·35-s − 1.97·37-s + 0.973·38-s + 0.158·40-s − 1.56·41-s − 1.21·43-s + 0.301·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.42065815219256, −13.09732529096473, −12.36313184295027, −11.93126105782880, −11.62696983234921, −11.27806041203045, −10.48174519611125, −10.10991274282009, −9.794592946652077, −9.074685237114602, −8.544518898181383, −8.155851837373144, −7.579159698115738, −6.849304222623158, −6.656856721452073, −6.033229497435603, −5.491497843627037, −5.015182161387565, −4.669408260935852, −3.739505587655113, −3.510069488830551, −2.978395307891694, −1.996769547139215, −1.698002610679760, −1.091834008476164, 0,
1.091834008476164, 1.698002610679760, 1.996769547139215, 2.978395307891694, 3.510069488830551, 3.739505587655113, 4.669408260935852, 5.015182161387565, 5.491497843627037, 6.033229497435603, 6.656856721452073, 6.849304222623158, 7.579159698115738, 8.155851837373144, 8.544518898181383, 9.074685237114602, 9.794592946652077, 10.10991274282009, 10.48174519611125, 11.27806041203045, 11.62696983234921, 11.93126105782880, 12.36313184295027, 13.09732529096473, 13.42065815219256
