| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s + 6·13-s − 14-s − 15-s + 16-s + 17-s + 18-s + 20-s + 21-s − 3·22-s + 23-s − 24-s − 4·25-s + 6·26-s − 27-s − 28-s + 8·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.92901088593321, −12.28575010878946, −11.95501466492428, −11.40813976311840, −11.10198360567617, −10.44695270016749, −10.22007893036651, −9.889465992455367, −9.084458611001842, −8.651313788086243, −8.164641343330704, −7.661096662209699, −7.075691590551192, −6.496782523804287, −6.258387452728759, −5.749172580269018, −5.344520103225724, −4.856591731297775, −4.330116654966627, −3.651475018786992, −3.290819852055563, −2.762778774945347, −1.971935904946769, −1.521937363427983, −0.8345036130558969, 0,
0.8345036130558969, 1.521937363427983, 1.971935904946769, 2.762778774945347, 3.290819852055563, 3.651475018786992, 4.330116654966627, 4.856591731297775, 5.344520103225724, 5.749172580269018, 6.258387452728759, 6.496782523804287, 7.075691590551192, 7.661096662209699, 8.164641343330704, 8.651313788086243, 9.084458611001842, 9.889465992455367, 10.22007893036651, 10.44695270016749, 11.10198360567617, 11.40813976311840, 11.95501466492428, 12.28575010878946, 12.92901088593321
