| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s − 2·10-s + 2·12-s − 13-s − 2·14-s + 15-s − 4·16-s + 4·17-s − 2·18-s + 19-s + 2·20-s + 21-s + 4·23-s + 25-s + 2·26-s + 27-s + 2·28-s − 3·29-s − 2·30-s + 9·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s − 16-s + 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.377·28-s − 0.557·29-s − 0.365·30-s + 1.61·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 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
−13.53276007220160, −13.18285357792860, −12.31565048045826, −12.07087237087984, −11.44337957589306, −10.91970448212398, −10.35437097664531, −10.07799038739231, −9.676320836574101, −9.077941490891189, −8.805950704369538, −8.139806402164505, −7.992195530762404, −7.226559424310891, −7.005039192530364, −6.383892031208011, −5.625467711064967, −5.083667065621857, −4.600007882084889, −3.883591244461703, −3.106349189124683, −2.704145372150283, −1.923171837536497, −1.435868872597221, −0.9395551657537346, 0,
0.9395551657537346, 1.435868872597221, 1.923171837536497, 2.704145372150283, 3.106349189124683, 3.883591244461703, 4.600007882084889, 5.083667065621857, 5.625467711064967, 6.383892031208011, 7.005039192530364, 7.226559424310891, 7.992195530762404, 8.139806402164505, 8.805950704369538, 9.077941490891189, 9.676320836574101, 10.07799038739231, 10.35437097664531, 10.91970448212398, 11.44337957589306, 12.07087237087984, 12.31565048045826, 13.18285357792860, 13.53276007220160
