| L(s) = 1 | − 1.88·2-s − 3.34·3-s + 1.57·4-s + 6.32·6-s − 7-s + 0.810·8-s + 8.18·9-s − 4.79·11-s − 5.25·12-s + 2.71·13-s + 1.88·14-s − 4.67·16-s − 0.520·17-s − 15.4·18-s + 0.405·19-s + 3.34·21-s + 9.05·22-s − 5.82·23-s − 2.71·24-s − 5.12·26-s − 17.3·27-s − 1.57·28-s + 0.163·29-s − 9.18·31-s + 7.21·32-s + 16.0·33-s + 0.982·34-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 1.93·3-s + 0.785·4-s + 2.58·6-s − 0.377·7-s + 0.286·8-s + 2.72·9-s − 1.44·11-s − 1.51·12-s + 0.752·13-s + 0.505·14-s − 1.16·16-s − 0.126·17-s − 3.64·18-s + 0.0930·19-s + 0.729·21-s + 1.93·22-s − 1.21·23-s − 0.553·24-s − 1.00·26-s − 3.34·27-s − 0.296·28-s + 0.0303·29-s − 1.65·31-s + 1.27·32-s + 2.78·33-s + 0.168·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1992820682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1992820682\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.88T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 0.520T + 17T^{2} \) |
| 19 | \( 1 - 0.405T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.163T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 1.98T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 - 8.65T + 61T^{2} \) |
| 67 | \( 1 - 7.72T + 67T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 + 2.38T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−10.20172685812857652861610185617, −9.661445478592356354744640819717, −8.466217233916646351978292952546, −7.52598349992473229230379840501, −6.84887778457395237877047382863, −5.84453625938192914854974818920, −5.17227656745360752254168327419, −3.99425352217126385964214022213, −1.87961127812172784989789306050, −0.47400405256328581197096774433,
0.47400405256328581197096774433, 1.87961127812172784989789306050, 3.99425352217126385964214022213, 5.17227656745360752254168327419, 5.84453625938192914854974818920, 6.84887778457395237877047382863, 7.52598349992473229230379840501, 8.466217233916646351978292952546, 9.661445478592356354744640819717, 10.20172685812857652861610185617
