| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 11-s + 2·13-s + 16-s − 17-s − 3·19-s + 2·20-s + 22-s + 23-s − 25-s − 2·26-s + 29-s − 2·31-s − 32-s + 34-s − 5·37-s + 3·38-s − 2·40-s + 10·41-s + 43-s − 44-s − 46-s − 7·47-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 + 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 0.447·20-s + 0.213·22-s + 0.208·23-s − 1/5·25-s − 0.392·26-s + 0.185·29-s − 0.359·31-s − 0.176·32-s + 0.171·34-s − 0.821·37-s + 0.486·38-s − 0.316·40-s + 1.56·41-s + 0.152·43-s − 0.150·44-s − 0.147·46-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.47025949539040111908408626278, −6.52583319483010552707774665565, −6.19257038013467596096264710291, −5.44598355032017029545060399064, −4.64975852135252741522105715146, −3.69424173386618508370048645432, −2.78908945837774141120478894029, −2.02214958164271635338498240195, −1.29860539333982514897792539671, 0,
1.29860539333982514897792539671, 2.02214958164271635338498240195, 2.78908945837774141120478894029, 3.69424173386618508370048645432, 4.64975852135252741522105715146, 5.44598355032017029545060399064, 6.19257038013467596096264710291, 6.52583319483010552707774665565, 7.47025949539040111908408626278
