| L(s) = 1 | + 2-s − 2.64·3-s + 4-s + 2.31·5-s − 2.64·6-s − 7-s + 8-s + 4.01·9-s + 2.31·10-s − 2.36·11-s − 2.64·12-s − 5.11·13-s − 14-s − 6.12·15-s + 16-s + 0.255·17-s + 4.01·18-s + 2.31·20-s + 2.64·21-s − 2.36·22-s + 1.44·23-s − 2.64·24-s + 0.354·25-s − 5.11·26-s − 2.68·27-s − 28-s + 0.240·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.03·5-s − 1.08·6-s − 0.377·7-s + 0.353·8-s + 1.33·9-s + 0.731·10-s − 0.712·11-s − 0.764·12-s − 1.41·13-s − 0.267·14-s − 1.58·15-s + 0.250·16-s + 0.0619·17-s + 0.946·18-s + 0.517·20-s + 0.577·21-s − 0.504·22-s + 0.301·23-s − 0.540·24-s + 0.0708·25-s − 1.00·26-s − 0.516·27-s − 0.188·28-s + 0.0446·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.626567760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.626567760\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 11 | \( 1 + 2.36T + 11T^{2} \) |
| 13 | \( 1 + 5.11T + 13T^{2} \) |
| 17 | \( 1 - 0.255T + 17T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 - 0.240T + 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.03T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.59T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 5.36T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 8.66T + 97T^{2} \) |
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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
−8.016259145957896605992907520112, −6.95521185245966194499617613640, −6.75998187470031561113791246528, −5.82689930695092702288229912975, −5.27060711147628066184399654487, −5.02355094765596189996738935485, −3.96675798029164922014219398873, −2.73922741519148864547414930255, −2.02396642044320514936180090490, −0.65312856014519352176763636431,
0.65312856014519352176763636431, 2.02396642044320514936180090490, 2.73922741519148864547414930255, 3.96675798029164922014219398873, 5.02355094765596189996738935485, 5.27060711147628066184399654487, 5.82689930695092702288229912975, 6.75998187470031561113791246528, 6.95521185245966194499617613640, 8.016259145957896605992907520112
