L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s + 6.69·13-s − 14-s + 16-s + 7.19·17-s − 1.84·19-s + 20-s − 22-s − 1.84·23-s + 25-s + 6.69·26-s − 28-s − 6.84·29-s + 6·31-s + 32-s + 7.19·34-s − 35-s − 6.54·37-s − 1.84·38-s + 40-s + 9.34·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.301·11-s + 1.85·13-s − 0.267·14-s + 0.250·16-s + 1.74·17-s − 0.424·19-s + 0.223·20-s − 0.213·22-s − 0.385·23-s + 0.200·25-s + 1.31·26-s − 0.188·28-s − 1.27·29-s + 1.07·31-s + 0.176·32-s + 1.23·34-s − 0.169·35-s − 1.07·37-s − 0.300·38-s + 0.158·40-s + 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.975800734\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.975800734\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 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
−7.929815149594036025196028559036, −7.13662932212886184525344086457, −6.29810442884946762081809377875, −5.81619057481646662644419545789, −5.34633783600358587387098007153, −4.22397055680046147324046886656, −3.59336878002083176972609905955, −2.94938794828874688873261728694, −1.87112572570495765603572695450, −0.964655340803238414798139798134,
0.964655340803238414798139798134, 1.87112572570495765603572695450, 2.94938794828874688873261728694, 3.59336878002083176972609905955, 4.22397055680046147324046886656, 5.34633783600358587387098007153, 5.81619057481646662644419545789, 6.29810442884946762081809377875, 7.13662932212886184525344086457, 7.929815149594036025196028559036