| L(s) = 1 | − 0.438·2-s + 5.12·3-s − 7.80·4-s + 6.31·5-s − 2.24·6-s + 6.93·8-s − 0.753·9-s − 2.76·10-s − 0.630·11-s − 40·12-s + 9.23·13-s + 32.3·15-s + 59.4·16-s + 58.4·17-s + 0.330·18-s − 1.30·19-s − 49.3·20-s + 0.276·22-s − 106.·23-s + 35.5·24-s − 85.1·25-s − 4.04·26-s − 142.·27-s − 284.·29-s − 14.1·30-s − 31·31-s − 81.5·32-s + ⋯ |
| L(s) = 1 | − 0.155·2-s + 0.985·3-s − 0.975·4-s + 0.564·5-s − 0.152·6-s + 0.306·8-s − 0.0279·9-s − 0.0875·10-s − 0.0172·11-s − 0.962·12-s + 0.196·13-s + 0.556·15-s + 0.928·16-s + 0.834·17-s + 0.00432·18-s − 0.0156·19-s − 0.551·20-s + 0.00267·22-s − 0.963·23-s + 0.301·24-s − 0.680·25-s − 0.0305·26-s − 1.01·27-s − 1.82·29-s − 0.0863·30-s − 0.179·31-s − 0.450·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 31 | \( 1 + 31T \) |
| good | 2 | \( 1 + 0.438T + 8T^{2} \) |
| 3 | \( 1 - 5.12T + 27T^{2} \) |
| 5 | \( 1 - 6.31T + 125T^{2} \) |
| 11 | \( 1 + 0.630T + 1.33e3T^{2} \) |
| 13 | \( 1 - 9.23T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.30T + 6.85e3T^{2} \) |
| 23 | \( 1 + 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 284.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 99.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 497.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 288.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 245.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 405.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 630.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 293.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 997.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 666.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−8.800889881247225597412283333435, −7.937332373924883591166391060266, −7.57060513039081482788036756564, −5.97041121005991378442694230215, −5.54651320557720090053289992455, −4.24046735495280036284404358503, −3.58449329495688175860940608854, −2.49608443400729120045691065618, −1.42986790643706709243949837051, 0,
1.42986790643706709243949837051, 2.49608443400729120045691065618, 3.58449329495688175860940608854, 4.24046735495280036284404358503, 5.54651320557720090053289992455, 5.97041121005991378442694230215, 7.57060513039081482788036756564, 7.937332373924883591166391060266, 8.800889881247225597412283333435
