| L(s) = 1 | + 2.73·2-s + 3.08·3-s + 5.49·4-s − 5-s + 8.43·6-s − 2.37·7-s + 9.55·8-s + 6.48·9-s − 2.73·10-s − 0.177·11-s + 16.9·12-s − 6.45·13-s − 6.49·14-s − 3.08·15-s + 15.1·16-s + 5.58·17-s + 17.7·18-s − 4.33·19-s − 5.49·20-s − 7.31·21-s − 0.486·22-s − 5.56·23-s + 29.4·24-s + 25-s − 17.6·26-s + 10.7·27-s − 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 1.77·3-s + 2.74·4-s − 0.447·5-s + 3.44·6-s − 0.896·7-s + 3.37·8-s + 2.16·9-s − 0.865·10-s − 0.0535·11-s + 4.88·12-s − 1.78·13-s − 1.73·14-s − 0.795·15-s + 3.79·16-s + 1.35·17-s + 4.18·18-s − 0.994·19-s − 1.22·20-s − 1.59·21-s − 0.103·22-s − 1.16·23-s + 6.00·24-s + 0.200·25-s − 3.46·26-s + 2.06·27-s − 2.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.069094055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.069094055\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 - 3.08T + 3T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 0.177T + 11T^{2} \) |
| 13 | \( 1 + 6.45T + 13T^{2} \) |
| 17 | \( 1 - 5.58T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 + 0.899T + 31T^{2} \) |
| 37 | \( 1 - 1.98T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 0.911T + 67T^{2} \) |
| 71 | \( 1 + 8.09T + 71T^{2} \) |
| 73 | \( 1 - 0.786T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 0.894T + 83T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−9.201533955323872334308364536136, −7.84191040657237863053592013258, −7.62353356378418639937212836335, −6.80452916660633512905746825061, −5.87869492480818087060945418460, −4.75043761579676401256703819801, −4.04714255260935879712849362182, −3.34315267451390834303281233318, −2.70778225235264855749418885905, −1.97368739633121493271136369024,
1.97368739633121493271136369024, 2.70778225235264855749418885905, 3.34315267451390834303281233318, 4.04714255260935879712849362182, 4.75043761579676401256703819801, 5.87869492480818087060945418460, 6.80452916660633512905746825061, 7.62353356378418639937212836335, 7.84191040657237863053592013258, 9.201533955323872334308364536136
