| L(s) = 1 | + 2·5-s + 4·11-s + 6·17-s + 8·19-s + 8·23-s − 25-s + 2·29-s + 8·31-s − 10·37-s + 6·41-s − 4·43-s − 7·49-s − 14·53-s + 8·55-s − 12·59-s + 10·61-s + 8·67-s + 14·73-s + 4·79-s + 4·83-s + 12·85-s − 10·89-s + 16·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 1.45·17-s + 1.83·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 49-s − 1.92·53-s + 1.07·55-s − 1.56·59-s + 1.28·61-s + 0.977·67-s + 1.63·73-s + 0.450·79-s + 0.439·83-s + 1.30·85-s − 1.05·89-s + 1.64·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.975550339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.975550339\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.95059707129017, −13.47594597467985, −12.66298596239658, −12.39033570645509, −11.83740653519900, −11.35590542926163, −10.92266822761746, −10.05133232283806, −9.856709740993381, −9.377779221728424, −9.027516452772536, −8.221699715084305, −7.814129877456102, −7.138411393225591, −6.626771777923364, −6.227417086130188, −5.513075051577050, −5.116669349494611, −4.658925614986483, −3.646148264118841, −3.272722241312738, −2.782124793897823, −1.770566128490455, −1.258448593301944, −0.7981109612370910,
0.7981109612370910, 1.258448593301944, 1.770566128490455, 2.782124793897823, 3.272722241312738, 3.646148264118841, 4.658925614986483, 5.116669349494611, 5.513075051577050, 6.227417086130188, 6.626771777923364, 7.138411393225591, 7.814129877456102, 8.221699715084305, 9.027516452772536, 9.377779221728424, 9.856709740993381, 10.05133232283806, 10.92266822761746, 11.35590542926163, 11.83740653519900, 12.39033570645509, 12.66298596239658, 13.47594597467985, 13.95059707129017
