| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 4·11-s − 6·13-s + 15-s − 6·17-s + 21-s + 4·23-s + 25-s + 27-s + 8·29-s − 2·31-s + 4·33-s + 35-s − 8·37-s − 6·39-s + 4·41-s + 4·43-s + 45-s + 6·47-s + 49-s − 6·51-s − 6·53-s + 4·55-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.258·15-s − 1.45·17-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.359·31-s + 0.696·33-s + 0.169·35-s − 1.31·37-s − 0.960·39-s + 0.624·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.093600624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.093600624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−16.05626252592544, −15.63101294079180, −14.93558170636351, −14.39852724726929, −14.19507031063523, −13.48764650098771, −12.85120610048834, −12.24106199250196, −11.80956213417082, −10.99995900144498, −10.47261476267464, −9.762756889684554, −9.136281826463441, −8.898390530238706, −8.116931207940712, −7.277515603630475, −6.867769274253276, −6.296376906488516, −5.269184407891883, −4.677670995236448, −4.160199206410978, −3.171321689684021, −2.389298464111646, −1.854451569020314, −0.7583078548309707,
0.7583078548309707, 1.854451569020314, 2.389298464111646, 3.171321689684021, 4.160199206410978, 4.677670995236448, 5.269184407891883, 6.296376906488516, 6.867769274253276, 7.277515603630475, 8.116931207940712, 8.898390530238706, 9.136281826463441, 9.762756889684554, 10.47261476267464, 10.99995900144498, 11.80956213417082, 12.24106199250196, 12.85120610048834, 13.48764650098771, 14.19507031063523, 14.39852724726929, 14.93558170636351, 15.63101294079180, 16.05626252592544
