| L(s) = 1 | − 2-s − 1.27·3-s + 4-s + 5-s + 1.27·6-s − 0.316·7-s − 8-s − 1.38·9-s − 10-s + 11-s − 1.27·12-s − 0.503·13-s + 0.316·14-s − 1.27·15-s + 16-s − 3.04·17-s + 1.38·18-s − 5.80·19-s + 20-s + 0.402·21-s − 22-s − 9.28·23-s + 1.27·24-s + 25-s + 0.503·26-s + 5.57·27-s − 0.316·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.734·3-s + 0.5·4-s + 0.447·5-s + 0.519·6-s − 0.119·7-s − 0.353·8-s − 0.461·9-s − 0.316·10-s + 0.301·11-s − 0.367·12-s − 0.139·13-s + 0.0846·14-s − 0.328·15-s + 0.250·16-s − 0.738·17-s + 0.326·18-s − 1.33·19-s + 0.223·20-s + 0.0878·21-s − 0.213·22-s − 1.93·23-s + 0.259·24-s + 0.200·25-s + 0.0987·26-s + 1.07·27-s − 0.0598·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6017912470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6017912470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.27T + 3T^{2} \) |
| 7 | \( 1 + 0.316T + 7T^{2} \) |
| 13 | \( 1 + 0.503T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 + 9.28T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 + 0.430T + 41T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 7.67T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 - 3.96T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−8.365387389076259954660078431493, −7.64828782540341752622031613993, −6.70518842801589280126819262724, −6.10182438443377551964237055453, −5.74994484105881943058746401153, −4.61963159655269873527047781058, −3.82527284589240715341461933895, −2.52277098840153535665104418258, −1.88947716941194334933834353847, −0.47341552674166497677161911719,
0.47341552674166497677161911719, 1.88947716941194334933834353847, 2.52277098840153535665104418258, 3.82527284589240715341461933895, 4.61963159655269873527047781058, 5.74994484105881943058746401153, 6.10182438443377551964237055453, 6.70518842801589280126819262724, 7.64828782540341752622031613993, 8.365387389076259954660078431493
