| L(s) = 1 | + 2-s − 3-s + 4-s + 1.02·5-s − 6-s − 3.83·7-s + 8-s + 9-s + 1.02·10-s + 4.68·11-s − 12-s + 13-s − 3.83·14-s − 1.02·15-s + 16-s − 4.11·17-s + 18-s − 0.252·19-s + 1.02·20-s + 3.83·21-s + 4.68·22-s − 2.89·23-s − 24-s − 3.94·25-s + 26-s − 27-s − 3.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.459·5-s − 0.408·6-s − 1.44·7-s + 0.353·8-s + 0.333·9-s + 0.324·10-s + 1.41·11-s − 0.288·12-s + 0.277·13-s − 1.02·14-s − 0.265·15-s + 0.250·16-s − 0.998·17-s + 0.235·18-s − 0.0579·19-s + 0.229·20-s + 0.837·21-s + 0.999·22-s − 0.603·23-s − 0.204·24-s − 0.789·25-s + 0.196·26-s − 0.192·27-s − 0.724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.474604849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.474604849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 4.68T + 11T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 + 0.252T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 7.44T + 29T^{2} \) |
| 31 | \( 1 + 0.252T + 31T^{2} \) |
| 37 | \( 1 - 8.61T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 2.97T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 + 9.34T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 3.32T + 83T^{2} \) |
| 89 | \( 1 - 8.76T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−7.54262198118665897157308929359, −6.57881204113114803663626476602, −6.49704653498696500227696331046, −5.97231565458607896141293914690, −5.08187090942374015541960514368, −4.12715261757106418641173896072, −3.76613955386939556854576477546, −2.76344956520029270509543242719, −1.87934732716211931515496881626, −0.71435626186233512978061217317,
0.71435626186233512978061217317, 1.87934732716211931515496881626, 2.76344956520029270509543242719, 3.76613955386939556854576477546, 4.12715261757106418641173896072, 5.08187090942374015541960514368, 5.97231565458607896141293914690, 6.49704653498696500227696331046, 6.57881204113114803663626476602, 7.54262198118665897157308929359
