| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s + 3·11-s + 13-s + 3·15-s − 17-s − 4·19-s + 3·21-s − 4·23-s + 25-s + 9·27-s − 9·29-s − 6·31-s + 9·33-s + 35-s − 8·37-s + 3·39-s + 6·41-s + 8·43-s + 6·45-s + 7·47-s + 49-s − 3·51-s − 8·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 0.904·11-s + 0.277·13-s + 0.774·15-s − 0.242·17-s − 0.917·19-s + 0.654·21-s − 0.834·23-s + 1/5·25-s + 1.73·27-s − 1.67·29-s − 1.07·31-s + 1.56·33-s + 0.169·35-s − 1.31·37-s + 0.480·39-s + 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.02·47-s + 1/7·49-s − 0.420·51-s − 1.09·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.368663974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.368663974\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−9.378517159078193926045921616149, −9.138540081877577152904672911411, −8.307879870828903559150374065212, −7.54899029587018689972345269630, −6.70036129479588863633569358046, −5.59641868623944705082898844853, −4.16990098847917098612516570173, −3.67651131106166908936746490743, −2.34543980751330993644717286952, −1.66007656338102233094959158350,
1.66007656338102233094959158350, 2.34543980751330993644717286952, 3.67651131106166908936746490743, 4.16990098847917098612516570173, 5.59641868623944705082898844853, 6.70036129479588863633569358046, 7.54899029587018689972345269630, 8.307879870828903559150374065212, 9.138540081877577152904672911411, 9.378517159078193926045921616149
