| L(s) = 1 | + 2-s − 2.04·3-s + 4-s + 1.23·5-s − 2.04·6-s − 7-s + 8-s + 1.19·9-s + 1.23·10-s + 6.18·11-s − 2.04·12-s + 5.70·13-s − 14-s − 2.53·15-s + 16-s − 4.82·17-s + 1.19·18-s + 1.23·20-s + 2.04·21-s + 6.18·22-s + 1.87·23-s − 2.04·24-s − 3.47·25-s + 5.70·26-s + 3.69·27-s − 28-s + 8.07·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.18·3-s + 0.5·4-s + 0.552·5-s − 0.836·6-s − 0.377·7-s + 0.353·8-s + 0.398·9-s + 0.390·10-s + 1.86·11-s − 0.591·12-s + 1.58·13-s − 0.267·14-s − 0.653·15-s + 0.250·16-s − 1.16·17-s + 0.281·18-s + 0.276·20-s + 0.446·21-s + 1.31·22-s + 0.391·23-s − 0.418·24-s − 0.694·25-s + 1.11·26-s + 0.711·27-s − 0.188·28-s + 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.617127684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.617127684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 - 6.18T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 + 8.59T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 3.74T + 47T^{2} \) |
| 53 | \( 1 + 1.03T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 + 6.88T + 67T^{2} \) |
| 71 | \( 1 - 4.86T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 2.96T + 83T^{2} \) |
| 89 | \( 1 - 5.79T + 89T^{2} \) |
| 97 | \( 1 + 6.97T + 97T^{2} \) |
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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
−8.402876007766116025639957059876, −6.87483522148290992784027133053, −6.51960014736850163519522116473, −6.19319595633810094963810936094, −5.46723935601142225917630307776, −4.54168233820779441390531523357, −3.95000122732971054554450252257, −3.04307342446471973962715158330, −1.74837309036462430194577469017, −0.899401471726897013759029234746,
0.899401471726897013759029234746, 1.74837309036462430194577469017, 3.04307342446471973962715158330, 3.95000122732971054554450252257, 4.54168233820779441390531523357, 5.46723935601142225917630307776, 6.19319595633810094963810936094, 6.51960014736850163519522116473, 6.87483522148290992784027133053, 8.402876007766116025639957059876
