| L(s) = 1 | − 2-s − 0.409·3-s + 4-s + 5-s + 0.409·6-s − 1.22·7-s − 8-s − 2.83·9-s − 10-s − 1.12·11-s − 0.409·12-s + 4.02·13-s + 1.22·14-s − 0.409·15-s + 16-s − 6.64·17-s + 2.83·18-s + 0.169·19-s + 20-s + 0.501·21-s + 1.12·22-s + 2.66·23-s + 0.409·24-s + 25-s − 4.02·26-s + 2.38·27-s − 1.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.236·3-s + 0.5·4-s + 0.447·5-s + 0.167·6-s − 0.463·7-s − 0.353·8-s − 0.944·9-s − 0.316·10-s − 0.340·11-s − 0.118·12-s + 1.11·13-s + 0.327·14-s − 0.105·15-s + 0.250·16-s − 1.61·17-s + 0.667·18-s + 0.0389·19-s + 0.223·20-s + 0.109·21-s + 0.240·22-s + 0.556·23-s + 0.0835·24-s + 0.200·25-s − 0.789·26-s + 0.459·27-s − 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8941487006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8941487006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 0.409T + 3T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 - 0.169T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 0.400T + 29T^{2} \) |
| 31 | \( 1 + 0.607T + 31T^{2} \) |
| 37 | \( 1 - 0.375T + 37T^{2} \) |
| 41 | \( 1 - 1.49T + 41T^{2} \) |
| 43 | \( 1 - 2.60T + 43T^{2} \) |
| 47 | \( 1 + 9.41T + 47T^{2} \) |
| 53 | \( 1 - 7.89T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 + 3.30T + 83T^{2} \) |
| 89 | \( 1 - 1.27T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−8.355464965230531029325611414977, −7.35205848047301012329021446463, −6.56318110299105805861475811411, −6.14397407964066357317429986092, −5.43418603541950539544415438033, −4.50687487783167244646278280633, −3.40453204459734332484003136362, −2.67847273987779016205207946859, −1.78394961895981494870644127676, −0.54501576529252256951473917443,
0.54501576529252256951473917443, 1.78394961895981494870644127676, 2.67847273987779016205207946859, 3.40453204459734332484003136362, 4.50687487783167244646278280633, 5.43418603541950539544415438033, 6.14397407964066357317429986092, 6.56318110299105805861475811411, 7.35205848047301012329021446463, 8.355464965230531029325611414977
