| L(s) = 1 | − 2-s + 3.34·3-s + 4-s − 5-s − 3.34·6-s − 7-s − 8-s + 8.19·9-s + 10-s + 11-s + 3.34·12-s + 6.69·13-s + 14-s − 3.34·15-s + 16-s − 7.19·17-s − 8.19·18-s − 1.84·19-s − 20-s − 3.34·21-s − 22-s + 1.84·23-s − 3.34·24-s + 25-s − 6.69·26-s + 17.3·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.93·3-s + 0.5·4-s − 0.447·5-s − 1.36·6-s − 0.377·7-s − 0.353·8-s + 2.73·9-s + 0.316·10-s + 0.301·11-s + 0.965·12-s + 1.85·13-s + 0.267·14-s − 0.863·15-s + 0.250·16-s − 1.74·17-s − 1.93·18-s − 0.424·19-s − 0.223·20-s − 0.730·21-s − 0.213·22-s + 0.385·23-s − 0.682·24-s + 0.200·25-s − 1.31·26-s + 3.34·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.097965460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097965460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 + 1.84T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 6.54T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 - 0.503T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 0.300T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 1.30T + 83T^{2} \) |
| 89 | \( 1 + 8.69T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−10.12931783416656873307310187531, −9.082367021117925585255363920823, −8.546578666809495227007706861792, −8.290384835808360788561045695506, −6.95938331858107350714800458445, −6.52701242703705687737240559725, −4.40760981648315973252342226898, −3.58299491889442570350027243061, −2.66362652996341208976806759276, −1.43465852026075341538500266859,
1.43465852026075341538500266859, 2.66362652996341208976806759276, 3.58299491889442570350027243061, 4.40760981648315973252342226898, 6.52701242703705687737240559725, 6.95938331858107350714800458445, 8.290384835808360788561045695506, 8.546578666809495227007706861792, 9.082367021117925585255363920823, 10.12931783416656873307310187531
