| L(s) = 1 | + 3-s + 7-s + 9-s − 2·11-s + 4·13-s − 2·17-s + 2·19-s + 21-s − 4·23-s + 27-s + 2·29-s − 6·31-s − 2·33-s − 6·37-s + 4·39-s + 6·41-s − 4·43-s + 49-s − 2·51-s + 8·53-s + 2·57-s + 10·61-s + 63-s − 12·67-s − 4·69-s − 14·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s − 0.485·17-s + 0.458·19-s + 0.218·21-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.348·33-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s + 1.09·53-s + 0.264·57-s + 1.28·61-s + 0.125·63-s − 1.46·67-s − 0.481·69-s − 1.66·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−15.20199319694616, −14.74880771435707, −14.14883743535062, −13.70813741520951, −13.21343478538963, −12.83386258800541, −12.00744688479242, −11.62980418043818, −10.91109455440225, −10.48888224384955, −9.984582816691781, −9.219491386256057, −8.728075455148159, −8.332805545678723, −7.690413000689201, −7.193623999428315, −6.539530729493839, −5.750039873300367, −5.373078124875318, −4.469248172309966, −3.968918376610647, −3.306568173213698, −2.597062406389267, −1.859040566243699, −1.188399477983350, 0,
1.188399477983350, 1.859040566243699, 2.597062406389267, 3.306568173213698, 3.968918376610647, 4.469248172309966, 5.373078124875318, 5.750039873300367, 6.539530729493839, 7.193623999428315, 7.690413000689201, 8.332805545678723, 8.728075455148159, 9.219491386256057, 9.984582816691781, 10.48888224384955, 10.91109455440225, 11.62980418043818, 12.00744688479242, 12.83386258800541, 13.21343478538963, 13.70813741520951, 14.14883743535062, 14.74880771435707, 15.20199319694616
