| L(s) = 1 | − 3.00·2-s + 3·3-s + 1.05·4-s + 17.7·5-s − 9.02·6-s + 14.5·7-s + 20.8·8-s + 9·9-s − 53.4·10-s − 2.09·11-s + 3.17·12-s − 1.54·13-s − 43.6·14-s + 53.3·15-s − 71.3·16-s + 76.3·17-s − 27.0·18-s + 105.·19-s + 18.8·20-s + 43.5·21-s + 6.29·22-s + 23·23-s + 62.6·24-s + 190.·25-s + 4.65·26-s + 27·27-s + 15.3·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 0.132·4-s + 1.58·5-s − 0.614·6-s + 0.783·7-s + 0.923·8-s + 0.333·9-s − 1.69·10-s − 0.0573·11-s + 0.0763·12-s − 0.0329·13-s − 0.833·14-s + 0.917·15-s − 1.11·16-s + 1.08·17-s − 0.354·18-s + 1.27·19-s + 0.210·20-s + 0.452·21-s + 0.0610·22-s + 0.208·23-s + 0.533·24-s + 1.52·25-s + 0.0351·26-s + 0.192·27-s + 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.685802418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.685802418\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 3.00T + 8T^{2} \) |
| 5 | \( 1 - 17.7T + 125T^{2} \) |
| 7 | \( 1 - 14.5T + 343T^{2} \) |
| 11 | \( 1 + 2.09T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.54T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 105.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 201.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 177.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 372.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 121.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 499.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 684.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 117.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 117.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 245.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 941.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 593.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 859.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 546.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 913.T + 9.12e5T^{2} \) |
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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
−8.982260461260831126692197071540, −8.106027733130618954300359642040, −7.57589221117237351347185746977, −6.67129365192480399933495848805, −5.44161680115429908434862747943, −5.06875094793913914324415420545, −3.70095004152789922884651649983, −2.46506907367745585141046950345, −1.62204454546331107416897647355, −0.964459758826083347507766264118,
0.964459758826083347507766264118, 1.62204454546331107416897647355, 2.46506907367745585141046950345, 3.70095004152789922884651649983, 5.06875094793913914324415420545, 5.44161680115429908434862747943, 6.67129365192480399933495848805, 7.57589221117237351347185746977, 8.106027733130618954300359642040, 8.982260461260831126692197071540
