| L(s) = 1 | + 3-s − 2·7-s + 9-s + 13-s − 2·17-s + 2·19-s − 2·21-s − 8·23-s + 27-s − 6·29-s + 2·31-s − 6·37-s + 39-s + 4·43-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s + 2·57-s − 4·59-s − 2·61-s − 2·63-s + 2·67-s − 8·69-s + 4·71-s + 2·73-s + 12·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.485·17-s + 0.458·19-s − 0.436·21-s − 1.66·23-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.986·37-s + 0.160·39-s + 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s + 0.264·57-s − 0.520·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s − 0.963·69-s + 0.474·71-s + 0.234·73-s + 1.35·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.460183798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.460183798\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.29281864933968, −13.58813404475601, −13.50405889885499, −12.67201722475518, −12.47643712938942, −11.72531796199055, −11.28827054686611, −10.61030984355992, −10.09836857782706, −9.621214712144807, −9.187900558738751, −8.680422763795228, −7.874360599762641, −7.781536746640034, −6.893304740819610, −6.425834038851542, −5.961509451992441, −5.226773111738353, −4.588137422678583, −3.811726805420007, −3.514653619199933, −2.811304427898241, −2.061959252458580, −1.525278395987496, −0.3778306030141561,
0.3778306030141561, 1.525278395987496, 2.061959252458580, 2.811304427898241, 3.514653619199933, 3.811726805420007, 4.588137422678583, 5.226773111738353, 5.961509451992441, 6.425834038851542, 6.893304740819610, 7.781536746640034, 7.874360599762641, 8.680422763795228, 9.187900558738751, 9.621214712144807, 10.09836857782706, 10.61030984355992, 11.28827054686611, 11.72531796199055, 12.47643712938942, 12.67201722475518, 13.50405889885499, 13.58813404475601, 14.29281864933968
