| L(s) = 1 | − 5-s − 7-s + 13-s + 17-s + 5·19-s − 3·23-s − 4·25-s + 3·29-s − 6·31-s + 35-s − 3·41-s + 3·43-s − 2·47-s + 49-s − 8·53-s + 14·59-s + 8·61-s − 65-s + 9·67-s + 3·71-s − 16·73-s − 4·79-s − 6·83-s − 85-s + 10·89-s − 91-s − 5·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.277·13-s + 0.242·17-s + 1.14·19-s − 0.625·23-s − 4/5·25-s + 0.557·29-s − 1.07·31-s + 0.169·35-s − 0.468·41-s + 0.457·43-s − 0.291·47-s + 1/7·49-s − 1.09·53-s + 1.82·59-s + 1.02·61-s − 0.124·65-s + 1.09·67-s + 0.356·71-s − 1.87·73-s − 0.450·79-s − 0.658·83-s − 0.108·85-s + 1.05·89-s − 0.104·91-s − 0.512·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.60234903936511, −14.28097292390624, −13.79089423450888, −13.07259348497943, −12.84262828552972, −12.07345650873870, −11.66130792582284, −11.33011466259364, −10.59137294636197, −9.997321599921714, −9.672834057959768, −9.033475111994412, −8.406727468645976, −7.934716360166866, −7.355901127532819, −6.895969042346821, −6.190882332374519, −5.614501775969563, −5.145681176988970, −4.318364546907708, −3.723291956007090, −3.310378746150011, −2.516473223394406, −1.730415521323817, −0.8946163174452279, 0,
0.8946163174452279, 1.730415521323817, 2.516473223394406, 3.310378746150011, 3.723291956007090, 4.318364546907708, 5.145681176988970, 5.614501775969563, 6.190882332374519, 6.895969042346821, 7.355901127532819, 7.934716360166866, 8.406727468645976, 9.033475111994412, 9.672834057959768, 9.997321599921714, 10.59137294636197, 11.33011466259364, 11.66130792582284, 12.07345650873870, 12.84262828552972, 13.07259348497943, 13.79089423450888, 14.28097292390624, 14.60234903936511
