| L(s) = 1 | − 7.81·3-s + 3.75·5-s − 31.3·7-s + 34.0·9-s + 11.5·11-s + 48.7·13-s − 29.3·15-s − 46.4·17-s − 77.9·19-s + 245.·21-s + 23·23-s − 110.·25-s − 54.8·27-s + 212.·29-s − 207.·31-s − 89.8·33-s − 117.·35-s − 197.·37-s − 380.·39-s − 383.·41-s − 437.·43-s + 127.·45-s + 156.·47-s + 642.·49-s + 362.·51-s − 12.0·53-s + 43.1·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 0.335·5-s − 1.69·7-s + 1.25·9-s + 0.315·11-s + 1.03·13-s − 0.504·15-s − 0.662·17-s − 0.940·19-s + 2.54·21-s + 0.208·23-s − 0.887·25-s − 0.390·27-s + 1.36·29-s − 1.20·31-s − 0.474·33-s − 0.568·35-s − 0.878·37-s − 1.56·39-s − 1.46·41-s − 1.55·43-s + 0.422·45-s + 0.484·47-s + 1.87·49-s + 0.995·51-s − 0.0311·53-s + 0.105·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4186893239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4186893239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 7.81T + 27T^{2} \) |
| 5 | \( 1 - 3.75T + 125T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 - 11.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 197.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 12.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 559.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 37.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 30.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 398.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 877.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 632.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 622.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 780.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 964.T + 9.12e5T^{2} \) |
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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
−9.230835804297417612702195818788, −8.536857258049946969045924264032, −7.02378220182373008668581803463, −6.42939238021668107673504958052, −6.11912177283485649586456308534, −5.18828726412000153136090200568, −4.10194290510178793733911784444, −3.19393518710012290360568154097, −1.68381061662655698593117081767, −0.33982794010532946786805569606,
0.33982794010532946786805569606, 1.68381061662655698593117081767, 3.19393518710012290360568154097, 4.10194290510178793733911784444, 5.18828726412000153136090200568, 6.11912177283485649586456308534, 6.42939238021668107673504958052, 7.02378220182373008668581803463, 8.536857258049946969045924264032, 9.230835804297417612702195818788
