| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 13-s + 16-s − 5·17-s + 8·19-s − 20-s − 22-s − 7·23-s − 4·25-s − 26-s + 2·29-s + 10·31-s − 32-s + 5·34-s − 10·37-s − 8·38-s + 40-s − 5·41-s − 10·43-s + 44-s + 7·46-s + 9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1/4·16-s − 1.21·17-s + 1.83·19-s − 0.223·20-s − 0.213·22-s − 1.45·23-s − 4/5·25-s − 0.196·26-s + 0.371·29-s + 1.79·31-s − 0.176·32-s + 0.857·34-s − 1.64·37-s − 1.29·38-s + 0.158·40-s − 0.780·41-s − 1.52·43-s + 0.150·44-s + 1.03·46-s + 1.31·47-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
| \(L(1)\) |
\(\approx\) |
\(0.8841020620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8841020620\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) | 1.5.b |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 17 T + p T^{2} \) | 1.83.r |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.52926656460123, −13.17539550539703, −12.23359349581530, −11.87503377583427, −11.65324991063634, −11.28209578281519, −10.29069238856922, −10.17664909541469, −9.787701418233086, −8.942950458854069, −8.631000744241233, −8.267206867615433, −7.504305356462339, −7.296148423022183, −6.568339155205505, −6.205212480983399, −5.531119950652633, −4.945695695201456, −4.267714651599205, −3.725125321388397, −3.171609297815288, −2.480486107022321, −1.792482442006870, −1.192218682142444, −0.3379747613977981,
0.3379747613977981, 1.192218682142444, 1.792482442006870, 2.480486107022321, 3.171609297815288, 3.725125321388397, 4.267714651599205, 4.945695695201456, 5.531119950652633, 6.205212480983399, 6.568339155205505, 7.296148423022183, 7.504305356462339, 8.267206867615433, 8.631000744241233, 8.942950458854069, 9.787701418233086, 10.17664909541469, 10.29069238856922, 11.28209578281519, 11.65324991063634, 11.87503377583427, 12.23359349581530, 13.17539550539703, 13.52926656460123
