| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 2·13-s − 2·14-s + 16-s − 5·25-s − 2·26-s + 2·28-s + 6·29-s − 32-s + 8·37-s + 41-s − 6·43-s − 3·49-s + 5·50-s + 2·52-s + 2·53-s − 2·56-s − 6·58-s + 8·59-s − 12·61-s + 64-s + 4·67-s − 4·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 25-s − 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.176·32-s + 1.31·37-s + 0.156·41-s − 0.914·43-s − 3/7·49-s + 0.707·50-s + 0.277·52-s + 0.274·53-s − 0.267·56-s − 0.787·58-s + 1.04·59-s − 1.53·61-s + 1/8·64-s + 0.488·67-s − 0.474·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.070217263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.070217263\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−12.20081185790646, −11.93346110435289, −11.48229871664628, −11.11403034571744, −10.56086787055573, −10.31443065418185, −9.627302475509772, −9.356933788781727, −8.803485919051547, −8.276365981879490, −7.921480077458811, −7.706985130766077, −6.967224026704199, −6.382954352983739, −6.243334246058172, −5.416912094929806, −5.110396869498690, −4.399576434216699, −3.989728688342036, −3.345063086368182, −2.744858452288567, −2.188299310907251, −1.614259138107595, −1.090048256431413, −0.4494827860124323,
0.4494827860124323, 1.090048256431413, 1.614259138107595, 2.188299310907251, 2.744858452288567, 3.345063086368182, 3.989728688342036, 4.399576434216699, 5.110396869498690, 5.416912094929806, 6.243334246058172, 6.382954352983739, 6.967224026704199, 7.706985130766077, 7.921480077458811, 8.276365981879490, 8.803485919051547, 9.356933788781727, 9.627302475509772, 10.31443065418185, 10.56086787055573, 11.11403034571744, 11.48229871664628, 11.93346110435289, 12.20081185790646
