| L(s) = 1 | + 2.13·2-s − 3-s + 2.54·4-s − 2.36·5-s − 2.13·6-s + 1.16·8-s + 9-s − 5.03·10-s + 4.44·11-s − 2.54·12-s + 0.290·13-s + 2.36·15-s − 2.61·16-s − 1.80·17-s + 2.13·18-s − 4.99·19-s − 6.00·20-s + 9.47·22-s − 1.55·23-s − 1.16·24-s + 0.572·25-s + 0.618·26-s − 27-s + 8.05·29-s + 5.03·30-s + 9.92·31-s − 7.89·32-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 0.577·3-s + 1.27·4-s − 1.05·5-s − 0.870·6-s + 0.410·8-s + 0.333·9-s − 1.59·10-s + 1.34·11-s − 0.734·12-s + 0.0804·13-s + 0.609·15-s − 0.653·16-s − 0.438·17-s + 0.502·18-s − 1.14·19-s − 1.34·20-s + 2.01·22-s − 0.324·23-s − 0.237·24-s + 0.114·25-s + 0.121·26-s − 0.192·27-s + 1.49·29-s + 0.918·30-s + 1.78·31-s − 1.39·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.941214644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941214644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 - 0.290T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 + 4.99T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 0.0265T + 47T^{2} \) |
| 53 | \( 1 + 2.59T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 6.61T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−8.042206430941960787432924117995, −6.78473709663558862367963858642, −6.61397987809242288339941061001, −5.98371223974699462118585465050, −4.92990726558615415489442842823, −4.34014354780171519378986822899, −4.01153160591705174465408157263, −3.15386427647088566787730419850, −2.12533603340466969126870806813, −0.73117296663903397898637741934,
0.73117296663903397898637741934, 2.12533603340466969126870806813, 3.15386427647088566787730419850, 4.01153160591705174465408157263, 4.34014354780171519378986822899, 4.92990726558615415489442842823, 5.98371223974699462118585465050, 6.61397987809242288339941061001, 6.78473709663558862367963858642, 8.042206430941960787432924117995
