| L(s) = 1 | + 2-s + 3-s + 4-s + 3·5-s + 6-s + 3·7-s + 8-s + 9-s + 3·10-s + 0.732·11-s + 12-s + 4.73·13-s + 3·14-s + 3·15-s + 16-s + 0.464·17-s + 18-s − 4.46·19-s + 3·20-s + 3·21-s + 0.732·22-s + 23-s + 24-s + 4·25-s + 4.73·26-s + 27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.34·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 0.333·9-s + 0.948·10-s + 0.220·11-s + 0.288·12-s + 1.31·13-s + 0.801·14-s + 0.774·15-s + 0.250·16-s + 0.112·17-s + 0.235·18-s − 1.02·19-s + 0.670·20-s + 0.654·21-s + 0.156·22-s + 0.208·23-s + 0.204·24-s + 0.800·25-s + 0.928·26-s + 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.893697174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.893697174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 9.39T + 47T^{2} \) |
| 53 | \( 1 - 2.53T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.521617222218064874829368939853, −7.79374147666536197674691367321, −6.70477771501253320983184293126, −6.24342437650037440482466507213, −5.38587185575190134227132702737, −4.75908611811333670646025170905, −3.84495065956566398078168293970, −2.98182890718561413265398533638, −1.75647714637458985883726461871, −1.62580224380410981962656552652,
1.62580224380410981962656552652, 1.75647714637458985883726461871, 2.98182890718561413265398533638, 3.84495065956566398078168293970, 4.75908611811333670646025170905, 5.38587185575190134227132702737, 6.24342437650037440482466507213, 6.70477771501253320983184293126, 7.79374147666536197674691367321, 8.521617222218064874829368939853
