| L(s) = 1 | − 1.46·2-s − 1.88·3-s + 0.143·4-s − 5-s + 2.76·6-s − 7-s + 2.71·8-s + 0.570·9-s + 1.46·10-s − 0.769·11-s − 0.271·12-s + 1.91·13-s + 1.46·14-s + 1.88·15-s − 4.26·16-s + 2.52·17-s − 0.835·18-s + 3.94·19-s − 0.143·20-s + 1.88·21-s + 1.12·22-s − 6.34·23-s − 5.13·24-s + 25-s − 2.80·26-s + 4.59·27-s − 0.143·28-s + ⋯ |
| L(s) = 1 | − 1.03·2-s − 1.09·3-s + 0.0719·4-s − 0.447·5-s + 1.12·6-s − 0.377·7-s + 0.960·8-s + 0.190·9-s + 0.463·10-s − 0.232·11-s − 0.0784·12-s + 0.532·13-s + 0.391·14-s + 0.487·15-s − 1.06·16-s + 0.613·17-s − 0.196·18-s + 0.905·19-s − 0.0321·20-s + 0.412·21-s + 0.240·22-s − 1.32·23-s − 1.04·24-s + 0.200·25-s − 0.550·26-s + 0.883·27-s − 0.0271·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4442309845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4442309845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 3 | \( 1 + 1.88T + 3T^{2} \) |
| 11 | \( 1 + 0.769T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 3.94T + 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 - 0.0132T + 29T^{2} \) |
| 31 | \( 1 + 1.12T + 31T^{2} \) |
| 37 | \( 1 - 0.322T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 7.51T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 4.86T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.956377442050998417211455975893, −7.21759504999749016959013871531, −6.59165511871027046151448572173, −5.65955498016177272021063231603, −5.31724811599032751888384827342, −4.27376433422389583797588259512, −3.65473674506698887467057246499, −2.48411648209563436951960924965, −1.20920853376217585478775547213, −0.48270765825997953312647487638,
0.48270765825997953312647487638, 1.20920853376217585478775547213, 2.48411648209563436951960924965, 3.65473674506698887467057246499, 4.27376433422389583797588259512, 5.31724811599032751888384827342, 5.65955498016177272021063231603, 6.59165511871027046151448572173, 7.21759504999749016959013871531, 7.956377442050998417211455975893
