| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 12-s − 4·13-s + 15-s + 16-s + 18-s + 8·19-s − 20-s − 22-s − 24-s + 25-s − 4·26-s − 27-s + 8·29-s + 30-s + 32-s + 33-s + 36-s + 8·37-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 − 0.301·11-s − 0.288·12-s − 1.10·13-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 1.48·29-s + 0.182·30-s + 0.176·32-s + 0.174·33-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−12.71403974928519, −12.42320937512589, −11.94526360926268, −11.60112517030134, −11.23323263525891, −10.72521330684268, −10.11904524277704, −9.770747686481343, −9.444290489212166, −8.610106992792990, −8.112071280894994, −7.626946792422559, −7.182044477205205, −6.897373501083937, −6.224059090531715, −5.693504255715750, −5.267136280573640, −4.836904854853910, −4.406230482145626, −3.853596386320756, −3.209545706065491, −2.671419288099712, −2.320585442220768, −1.248417850471593, −0.8847711813111889, 0,
0.8847711813111889, 1.248417850471593, 2.320585442220768, 2.671419288099712, 3.209545706065491, 3.853596386320756, 4.406230482145626, 4.836904854853910, 5.267136280573640, 5.693504255715750, 6.224059090531715, 6.897373501083937, 7.182044477205205, 7.626946792422559, 8.112071280894994, 8.610106992792990, 9.444290489212166, 9.770747686481343, 10.11904524277704, 10.72521330684268, 11.23323263525891, 11.60112517030134, 11.94526360926268, 12.42320937512589, 12.71403974928519
