| L(s) = 1 | + 2-s + 0.618·3-s + 4-s + 0.618·6-s + 7-s + 8-s − 2.61·9-s − 0.854·11-s + 0.618·12-s − 4·13-s + 14-s + 16-s + 17-s − 2.61·18-s − 3.61·19-s + 0.618·21-s − 0.854·22-s + 5.09·23-s + 0.618·24-s − 4·26-s − 3.47·27-s + 28-s − 8.61·29-s − 0.381·31-s + 32-s − 0.527·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s + 0.252·6-s + 0.377·7-s + 0.353·8-s − 0.872·9-s − 0.257·11-s + 0.178·12-s − 1.10·13-s + 0.267·14-s + 0.250·16-s + 0.242·17-s − 0.617·18-s − 0.830·19-s + 0.134·21-s − 0.182·22-s + 1.06·23-s + 0.126·24-s − 0.784·26-s − 0.668·27-s + 0.188·28-s − 1.60·29-s − 0.0686·31-s + 0.176·32-s − 0.0918·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 11 | \( 1 + 0.854T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 + 0.381T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 0.618T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 8.76T + 97T^{2} \) |
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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
−7.68025596717138404569502714597, −7.10056745934665870772245469520, −6.14432335211873751152391847804, −5.50902965764064822626388379720, −4.84763838987341718571065117546, −4.11437928225405586276453462636, −3.11679292363264095560894688719, −2.56253092833570864268856129388, −1.65187705114761134466444637340, 0,
1.65187705114761134466444637340, 2.56253092833570864268856129388, 3.11679292363264095560894688719, 4.11437928225405586276453462636, 4.84763838987341718571065117546, 5.50902965764064822626388379720, 6.14432335211873751152391847804, 7.10056745934665870772245469520, 7.68025596717138404569502714597
