| L(s) = 1 | + 2·3-s + 7-s + 9-s − 2·11-s + 5·13-s + 7·17-s − 6·19-s + 2·21-s + 3·23-s − 5·25-s − 4·27-s − 6·29-s − 4·31-s − 4·33-s − 3·37-s + 10·39-s − 8·41-s + 7·43-s − 8·47-s − 6·49-s + 14·51-s − 12·57-s − 5·59-s − 2·61-s + 63-s + 6·69-s − 14·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.38·13-s + 1.69·17-s − 1.37·19-s + 0.436·21-s + 0.625·23-s − 25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.696·33-s − 0.493·37-s + 1.60·39-s − 1.24·41-s + 1.06·43-s − 1.16·47-s − 6/7·49-s + 1.96·51-s − 1.58·57-s − 0.650·59-s − 0.256·61-s + 0.125·63-s + 0.722·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 157 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.76605195284047, −15.16248421416661, −14.90814082829662, −14.28277456510472, −13.82928292521365, −13.23136938223257, −12.89941405243315, −12.17411520503498, −11.43473773799457, −10.91463090040303, −10.41407431453194, −9.666533958803141, −9.147392722025643, −8.543359278910312, −8.034744835258991, −7.755832343634641, −6.957859555584826, −6.055146889193437, −5.618549610295501, −4.861176795624760, −3.855546344554549, −3.536655031757297, −2.860620308261707, −1.918814226079453, −1.436308980906796, 0,
1.436308980906796, 1.918814226079453, 2.860620308261707, 3.536655031757297, 3.855546344554549, 4.861176795624760, 5.618549610295501, 6.055146889193437, 6.957859555584826, 7.755832343634641, 8.034744835258991, 8.543359278910312, 9.147392722025643, 9.666533958803141, 10.41407431453194, 10.91463090040303, 11.43473773799457, 12.17411520503498, 12.89941405243315, 13.23136938223257, 13.82928292521365, 14.28277456510472, 14.90814082829662, 15.16248421416661, 15.76605195284047
