| L(s) = 1 | + 0.310·2-s + 3-s − 1.90·4-s + 5-s + 0.310·6-s + 1.00·7-s − 1.21·8-s + 9-s + 0.310·10-s + 2.90·11-s − 1.90·12-s + 0.662·13-s + 0.312·14-s + 15-s + 3.43·16-s + 4.30·17-s + 0.310·18-s − 1.26·19-s − 1.90·20-s + 1.00·21-s + 0.901·22-s + 7.23·23-s − 1.21·24-s + 25-s + 0.205·26-s + 27-s − 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.219·2-s + 0.577·3-s − 0.951·4-s + 0.447·5-s + 0.126·6-s + 0.380·7-s − 0.428·8-s + 0.333·9-s + 0.0982·10-s + 0.875·11-s − 0.549·12-s + 0.183·13-s + 0.0835·14-s + 0.258·15-s + 0.857·16-s + 1.04·17-s + 0.0731·18-s − 0.290·19-s − 0.425·20-s + 0.219·21-s + 0.192·22-s + 1.50·23-s − 0.247·24-s + 0.200·25-s + 0.0403·26-s + 0.192·27-s − 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.901891513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.901891513\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 0.310T + 2T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 - 0.662T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 6.78T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 41 | \( 1 + 9.54T + 41T^{2} \) |
| 43 | \( 1 + 0.00931T + 43T^{2} \) |
| 47 | \( 1 - 3.95T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + 8.08T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.245220840092001933716013086048, −7.45093139742188544905118198584, −6.67488002142703776315550131367, −5.78675279790239261669020844934, −5.15017505781324093626083255927, −4.41924614367895422982503188448, −3.62800163735932976872248589477, −3.01708830272710856930881696537, −1.76367124929755016624181391466, −0.913249742563974422030064883772,
0.913249742563974422030064883772, 1.76367124929755016624181391466, 3.01708830272710856930881696537, 3.62800163735932976872248589477, 4.41924614367895422982503188448, 5.15017505781324093626083255927, 5.78675279790239261669020844934, 6.67488002142703776315550131367, 7.45093139742188544905118198584, 8.245220840092001933716013086048
