| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 6·11-s − 2·14-s + 16-s − 2·17-s + 20-s − 6·22-s − 4·23-s + 25-s + 2·28-s − 8·29-s + 8·31-s − 32-s + 2·34-s + 2·35-s + 4·37-s − 40-s − 4·41-s − 6·43-s + 6·44-s + 4·46-s + 12·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s + 1.80·11-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.223·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s + 0.377·28-s − 1.48·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.338·35-s + 0.657·37-s − 0.158·40-s − 0.624·41-s − 0.914·43-s + 0.904·44-s + 0.589·46-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.317110595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.317110595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−15.12090213309211, −14.54795722724331, −13.96145140476509, −13.71076899527555, −12.84072656954699, −12.24696126144914, −11.63136350069757, −11.40794543080539, −10.82786519370841, −9.942195427221269, −9.807648899190607, −8.977573927404323, −8.714802853858999, −8.070213452298670, −7.423658995518015, −6.796194187028028, −6.309220029541605, −5.767296689237919, −4.992731894282618, −4.185380391717004, −3.782433530019151, −2.759258870181791, −1.944916743445002, −1.503916226413878, −0.6707310597711780,
0.6707310597711780, 1.503916226413878, 1.944916743445002, 2.759258870181791, 3.782433530019151, 4.185380391717004, 4.992731894282618, 5.767296689237919, 6.309220029541605, 6.796194187028028, 7.423658995518015, 8.070213452298670, 8.714802853858999, 8.977573927404323, 9.807648899190607, 9.942195427221269, 10.82786519370841, 11.40794543080539, 11.63136350069757, 12.24696126144914, 12.84072656954699, 13.71076899527555, 13.96145140476509, 14.54795722724331, 15.12090213309211
