| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 5·7-s + 8-s − 2·9-s − 10-s − 11-s + 12-s + 4·13-s + 5·14-s − 15-s + 16-s − 2·18-s + 6·19-s − 20-s + 5·21-s − 22-s + 4·23-s + 24-s + 25-s + 4·26-s − 5·27-s + 5·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.10·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.471·18-s + 1.37·19-s − 0.223·20-s + 1.09·21-s − 0.213·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.962·27-s + 0.944·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.928046438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.928046438\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 419 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.41618761763115, −14.03758152650179, −13.86357228757164, −13.32676735925713, −12.46004455980647, −11.98055253370226, −11.57574586101246, −11.02309849225709, −10.78956142043247, −10.10904695500106, −8.935313241608828, −8.865798555590996, −8.239362821557315, −7.654188799716210, −7.391062620627986, −6.590941726417708, −5.626969832437000, −5.426472135315580, −4.786630294066620, −4.167036270903004, −3.528769690769624, −2.952399221675287, −2.283571366751560, −1.470896450550363, −0.8681819064941723,
0.8681819064941723, 1.470896450550363, 2.283571366751560, 2.952399221675287, 3.528769690769624, 4.167036270903004, 4.786630294066620, 5.426472135315580, 5.626969832437000, 6.590941726417708, 7.391062620627986, 7.654188799716210, 8.239362821557315, 8.865798555590996, 8.935313241608828, 10.10904695500106, 10.78956142043247, 11.02309849225709, 11.57574586101246, 11.98055253370226, 12.46004455980647, 13.32676735925713, 13.86357228757164, 14.03758152650179, 14.41618761763115
