| L(s) = 1 | + 2-s + 4-s − 5-s − 2.37·7-s + 8-s − 10-s + 11-s + 2·13-s − 2.37·14-s + 16-s + 4.37·17-s + 6.37·19-s − 20-s + 22-s + 8.74·23-s + 25-s + 2·26-s − 2.37·28-s + 4.37·29-s − 2.37·31-s + 32-s + 4.37·34-s + 2.37·35-s + 3.62·37-s + 6.37·38-s − 40-s − 11.4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.896·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s − 0.634·14-s + 0.250·16-s + 1.06·17-s + 1.46·19-s − 0.223·20-s + 0.213·22-s + 1.82·23-s + 0.200·25-s + 0.392·26-s − 0.448·28-s + 0.811·29-s − 0.426·31-s + 0.176·32-s + 0.749·34-s + 0.400·35-s + 0.596·37-s + 1.03·38-s − 0.158·40-s − 1.79·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.311156842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.311156842\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 8.74T + 23T^{2} \) |
| 29 | \( 1 - 4.37T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 - 3.62T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8.74T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−9.987167497693696907981490082249, −9.274576879196523008943501056100, −8.215677478566040724031925613344, −7.24890440236121834873346396090, −6.59502373629883656343250751602, −5.59914590068054735675065520320, −4.75387261308194036035165751564, −3.42710888140596097956805943737, −3.10831468308433265340113329022, −1.16615174006350466134066828480,
1.16615174006350466134066828480, 3.10831468308433265340113329022, 3.42710888140596097956805943737, 4.75387261308194036035165751564, 5.59914590068054735675065520320, 6.59502373629883656343250751602, 7.24890440236121834873346396090, 8.215677478566040724031925613344, 9.274576879196523008943501056100, 9.987167497693696907981490082249
