| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 2·11-s + 12-s + 13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s + 20-s − 4·21-s + 2·22-s − 2·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.426·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 71 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.39529506223618, −14.72906153470456, −14.59128369481274, −13.79588393911185, −13.31215252333877, −13.06647056048184, −12.34242955772815, −12.07158997826576, −11.28736599883441, −10.49694101568091, −10.03856579406947, −9.643798039853824, −9.006559064266395, −8.456838649684964, −7.661599900989621, −7.029846452203152, −6.566156701831359, −5.868508404667448, −5.618816926077091, −4.558905861361606, −3.891118084987639, −3.342216242978052, −2.948776983231591, −1.999005595437974, −1.331762012225060, 0,
1.331762012225060, 1.999005595437974, 2.948776983231591, 3.342216242978052, 3.891118084987639, 4.558905861361606, 5.618816926077091, 5.868508404667448, 6.566156701831359, 7.029846452203152, 7.661599900989621, 8.456838649684964, 9.006559064266395, 9.643798039853824, 10.03856579406947, 10.49694101568091, 11.28736599883441, 12.07158997826576, 12.34242955772815, 13.06647056048184, 13.31215252333877, 13.79588393911185, 14.59128369481274, 14.72906153470456, 15.39529506223618
