| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 11-s − 13-s + 16-s + 8·17-s − 2·20-s + 22-s − 2·23-s − 25-s − 26-s + 6·29-s + 2·31-s + 32-s + 8·34-s + 10·37-s − 2·40-s − 2·41-s + 6·43-s + 44-s − 2·46-s − 8·47-s − 50-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 0.301·11-s − 0.277·13-s + 1/4·16-s + 1.94·17-s − 0.447·20-s + 0.213·22-s − 0.417·23-s − 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.37·34-s + 1.64·37-s − 0.316·40-s − 0.312·41-s + 0.914·43-s + 0.150·44-s − 0.294·46-s − 1.16·47-s − 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 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
−13.80940652127766, −13.25749787519327, −12.67087138129862, −12.30587018919451, −11.88825982614590, −11.50004184991610, −11.11318914826018, −10.32386014110005, −9.914545426394930, −9.613440289440527, −8.705003139735090, −8.202373003498610, −7.749964763053837, −7.446562078674682, −6.769859901988434, −6.191187248822924, −5.719299848370999, −5.185545816419254, −4.449422617554312, −4.179983655769020, −3.518769126514152, −2.989058025807291, −2.529938012858648, −1.480475829375751, −1.003394893067746, 0,
1.003394893067746, 1.480475829375751, 2.529938012858648, 2.989058025807291, 3.518769126514152, 4.179983655769020, 4.449422617554312, 5.185545816419254, 5.719299848370999, 6.191187248822924, 6.769859901988434, 7.446562078674682, 7.749964763053837, 8.202373003498610, 8.705003139735090, 9.613440289440527, 9.914545426394930, 10.32386014110005, 11.11318914826018, 11.50004184991610, 11.88825982614590, 12.30587018919451, 12.67087138129862, 13.25749787519327, 13.80940652127766
