| L(s) = 1 | − 2.78·2-s + 3-s + 5.78·4-s − 0.430·5-s − 2.78·6-s + 7-s − 10.5·8-s + 9-s + 1.20·10-s − 4.67·11-s + 5.78·12-s + 4.61·13-s − 2.78·14-s − 0.430·15-s + 17.8·16-s + 5.52·17-s − 2.78·18-s − 2.49·20-s + 21-s + 13.0·22-s − 1.98·23-s − 10.5·24-s − 4.81·25-s − 12.8·26-s + 27-s + 5.78·28-s − 6.41·29-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.577·3-s + 2.89·4-s − 0.192·5-s − 1.13·6-s + 0.377·7-s − 3.72·8-s + 0.333·9-s + 0.380·10-s − 1.40·11-s + 1.66·12-s + 1.27·13-s − 0.745·14-s − 0.111·15-s + 4.46·16-s + 1.33·17-s − 0.657·18-s − 0.557·20-s + 0.218·21-s + 2.77·22-s − 0.413·23-s − 2.15·24-s − 0.962·25-s − 2.52·26-s + 0.192·27-s + 1.09·28-s − 1.19·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7581 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9882011046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9882011046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 5 | \( 1 + 0.430T + 5T^{2} \) |
| 11 | \( 1 + 4.67T + 11T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 - 0.487T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 0.358T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 2.40T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 3.22T + 97T^{2} \) |
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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
−8.023658275420515524088820610319, −7.64760624976741389103136464849, −6.92920499067228837633941384469, −5.90829601870201104322759564402, −5.53430529683176933979843754891, −3.97263465568242541579291356637, −3.13231418226514475838773813202, −2.40433078652648909510496114157, −1.57286477329095711094179835687, −0.66937907867223875519700999487,
0.66937907867223875519700999487, 1.57286477329095711094179835687, 2.40433078652648909510496114157, 3.13231418226514475838773813202, 3.97263465568242541579291356637, 5.53430529683176933979843754891, 5.90829601870201104322759564402, 6.92920499067228837633941384469, 7.64760624976741389103136464849, 8.023658275420515524088820610319
