| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 13-s − 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 4·22-s + 24-s + 25-s − 26-s + 27-s − 2·29-s − 30-s − 8·31-s + 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.749931581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.749931581\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.64752326893849, −14.97718127181737, −14.54653087029467, −14.27709905187781, −13.55937964602609, −13.00601476067365, −12.52728999766030, −11.90355216761641, −11.34726525272716, −11.02044004947277, −10.07744438663006, −9.508498124054904, −8.989385940976213, −8.395447331582582, −7.513245000656431, −7.246046582415252, −6.588020008502473, −5.799745731886453, −5.229342234536668, −4.202460504859199, −4.057543322686475, −3.263037207888381, −2.540180579976920, −1.730912388671990, −0.8053103263001221,
0.8053103263001221, 1.730912388671990, 2.540180579976920, 3.263037207888381, 4.057543322686475, 4.202460504859199, 5.229342234536668, 5.799745731886453, 6.588020008502473, 7.246046582415252, 7.513245000656431, 8.395447331582582, 8.989385940976213, 9.508498124054904, 10.07744438663006, 11.02044004947277, 11.34726525272716, 11.90355216761641, 12.52728999766030, 13.00601476067365, 13.55937964602609, 14.27709905187781, 14.54653087029467, 14.97718127181737, 15.64752326893849
