| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s + 11-s − 2·13-s − 2·15-s − 6·17-s − 8·19-s − 4·21-s − 25-s + 27-s − 2·29-s + 33-s + 8·35-s + 37-s − 2·39-s − 6·41-s − 8·43-s − 2·45-s − 12·47-s + 9·49-s − 6·51-s − 6·53-s − 2·55-s − 8·57-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.174·33-s + 1.35·35-s + 0.164·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.75·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.269·55-s − 1.05·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−16.11757890106341, −15.58779675477190, −15.17501055660213, −14.84298838820907, −14.05052015372959, −13.30329651739275, −13.01577199254301, −12.56460614597038, −11.89571726317401, −11.29490814121481, −10.69204083749421, −9.984871709662782, −9.562314504696403, −8.886444149853542, −8.429005116054379, −7.838322017755982, −7.012869680447267, −6.558620654247160, −6.245888922560899, −5.026171819880218, −4.357832404463928, −3.807361351614071, −3.241577334098866, −2.472518349994483, −1.744856222563909, 0, 0,
1.744856222563909, 2.472518349994483, 3.241577334098866, 3.807361351614071, 4.357832404463928, 5.026171819880218, 6.245888922560899, 6.558620654247160, 7.012869680447267, 7.838322017755982, 8.429005116054379, 8.886444149853542, 9.562314504696403, 9.984871709662782, 10.69204083749421, 11.29490814121481, 11.89571726317401, 12.56460614597038, 13.01577199254301, 13.30329651739275, 14.05052015372959, 14.84298838820907, 15.17501055660213, 15.58779675477190, 16.11757890106341
