| L(s) = 1 | + 1.65·2-s − 8.44·3-s − 5.26·4-s − 2.08·5-s − 13.9·6-s − 21.9·8-s + 44.3·9-s − 3.44·10-s + 51.0·11-s + 44.5·12-s + 66.2·13-s + 17.5·15-s + 5.91·16-s + 80.6·17-s + 73.2·18-s + 8.13·19-s + 10.9·20-s + 84.3·22-s − 110.·23-s + 185.·24-s − 120.·25-s + 109.·26-s − 146.·27-s + 271.·29-s + 29.0·30-s + 257.·31-s + 185.·32-s + ⋯ |
| L(s) = 1 | + 0.584·2-s − 1.62·3-s − 0.658·4-s − 0.186·5-s − 0.949·6-s − 0.969·8-s + 1.64·9-s − 0.108·10-s + 1.39·11-s + 1.07·12-s + 1.41·13-s + 0.302·15-s + 0.0923·16-s + 1.15·17-s + 0.959·18-s + 0.0982·19-s + 0.122·20-s + 0.817·22-s − 0.998·23-s + 1.57·24-s − 0.965·25-s + 0.825·26-s − 1.04·27-s + 1.74·29-s + 0.176·30-s + 1.49·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.459659602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.459659602\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - 41T \) |
| good | 2 | \( 1 - 1.65T + 8T^{2} \) |
| 3 | \( 1 + 8.44T + 27T^{2} \) |
| 5 | \( 1 + 2.08T + 125T^{2} \) |
| 11 | \( 1 - 51.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 8.13T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 257.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 294.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 79.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 414.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 144.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 779.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 104.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 281.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 499.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.26e3T + 9.12e5T^{2} \) |
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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
−8.720227243537401964850376649733, −8.106563376185120810729409015220, −6.70634300780393032452737958155, −6.23774397162075296371657945706, −5.66440265875740372391353005183, −4.79390278979075438600829458206, −4.04992693641353113388669847592, −3.39871187014940499941185459818, −1.36282163875557193517534231080, −0.64166843933405945310096911945,
0.64166843933405945310096911945, 1.36282163875557193517534231080, 3.39871187014940499941185459818, 4.04992693641353113388669847592, 4.79390278979075438600829458206, 5.66440265875740372391353005183, 6.23774397162075296371657945706, 6.70634300780393032452737958155, 8.106563376185120810729409015220, 8.720227243537401964850376649733
