| L(s) = 1 | − 2-s − 3-s + 4-s − 0.503·5-s + 6-s + 2.75·7-s − 8-s + 9-s + 0.503·10-s + 6.03·11-s − 12-s − 13-s − 2.75·14-s + 0.503·15-s + 16-s − 6.29·17-s − 18-s − 6.31·19-s − 0.503·20-s − 2.75·21-s − 6.03·22-s − 7.22·23-s + 24-s − 4.74·25-s + 26-s − 27-s + 2.75·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.225·5-s + 0.408·6-s + 1.04·7-s − 0.353·8-s + 0.333·9-s + 0.159·10-s + 1.81·11-s − 0.288·12-s − 0.277·13-s − 0.737·14-s + 0.130·15-s + 0.250·16-s − 1.52·17-s − 0.235·18-s − 1.44·19-s − 0.112·20-s − 0.602·21-s − 1.28·22-s − 1.50·23-s + 0.204·24-s − 0.949·25-s + 0.196·26-s − 0.192·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.083184612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083184612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 0.503T + 5T^{2} \) |
| 7 | \( 1 - 2.75T + 7T^{2} \) |
| 11 | \( 1 - 6.03T + 11T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 6.31T + 19T^{2} \) |
| 23 | \( 1 + 7.22T + 23T^{2} \) |
| 29 | \( 1 - 9.35T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 + 6.99T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 + 9.24T + 59T^{2} \) |
| 61 | \( 1 + 0.178T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 4.54T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 0.865T + 83T^{2} \) |
| 89 | \( 1 - 0.262T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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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
−8.095712576564663723684061107658, −7.04697141809968252732503111253, −6.37235307163411462265412626093, −6.21573958254799578692433332498, −4.84894299533633670776498373784, −4.37641527331804506052429887676, −3.71269688490918243605167461290, −2.17250319558394160532226284575, −1.75934998831297184820209086181, −0.59389553833442049994122446076,
0.59389553833442049994122446076, 1.75934998831297184820209086181, 2.17250319558394160532226284575, 3.71269688490918243605167461290, 4.37641527331804506052429887676, 4.84894299533633670776498373784, 6.21573958254799578692433332498, 6.37235307163411462265412626093, 7.04697141809968252732503111253, 8.095712576564663723684061107658
