| L(s) = 1 | − 3-s − 7-s + 9-s + 2·11-s − 5·13-s + 3·17-s + 8·19-s + 21-s + 23-s − 27-s − 29-s + 9·31-s − 2·33-s − 2·37-s + 5·39-s + 7·41-s + 7·43-s − 12·47-s + 49-s − 3·51-s − 13·53-s − 8·57-s − 15·59-s − 7·61-s − 63-s − 4·67-s − 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.38·13-s + 0.727·17-s + 1.83·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.185·29-s + 1.61·31-s − 0.348·33-s − 0.328·37-s + 0.800·39-s + 1.09·41-s + 1.06·43-s − 1.75·47-s + 1/7·49-s − 0.420·51-s − 1.78·53-s − 1.05·57-s − 1.95·59-s − 0.896·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−15.42806574044010, −14.63441136768074, −14.19412118874293, −13.81353009548777, −13.06678327231910, −12.41648908439758, −12.13776145822244, −11.66586748170469, −11.13009754287448, −10.36462713656362, −9.917884406307152, −9.373348731283880, −9.147711587651715, −7.896068433084029, −7.700455318289663, −7.100725447878915, −6.353156373678302, −5.987551516290684, −5.128864923991004, −4.805880787178083, −4.081587982526987, −3.071897953594424, −2.888594701051244, −1.634353714876826, −0.9968311635016983, 0,
0.9968311635016983, 1.634353714876826, 2.888594701051244, 3.071897953594424, 4.081587982526987, 4.805880787178083, 5.128864923991004, 5.987551516290684, 6.353156373678302, 7.100725447878915, 7.700455318289663, 7.896068433084029, 9.147711587651715, 9.373348731283880, 9.917884406307152, 10.36462713656362, 11.13009754287448, 11.66586748170469, 12.13776145822244, 12.41648908439758, 13.06678327231910, 13.81353009548777, 14.19412118874293, 14.63441136768074, 15.42806574044010
