| L(s) = 1 | − 2-s − 3.17·3-s + 4-s + 4.25·5-s + 3.17·6-s + 4.60·7-s − 8-s + 7.06·9-s − 4.25·10-s − 1.37·11-s − 3.17·12-s + 5.86·13-s − 4.60·14-s − 13.4·15-s + 16-s − 2.42·17-s − 7.06·18-s − 0.898·19-s + 4.25·20-s − 14.6·21-s + 1.37·22-s − 6.07·23-s + 3.17·24-s + 13.0·25-s − 5.86·26-s − 12.8·27-s + 4.60·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.83·3-s + 0.5·4-s + 1.90·5-s + 1.29·6-s + 1.74·7-s − 0.353·8-s + 2.35·9-s − 1.34·10-s − 0.415·11-s − 0.915·12-s + 1.62·13-s − 1.23·14-s − 3.48·15-s + 0.250·16-s − 0.589·17-s − 1.66·18-s − 0.206·19-s + 0.951·20-s − 3.19·21-s + 0.293·22-s − 1.26·23-s + 0.647·24-s + 2.61·25-s − 1.15·26-s − 2.48·27-s + 0.871·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.546028578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546028578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 - 5.86T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 + 0.898T + 19T^{2} \) |
| 23 | \( 1 + 6.07T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 1.97T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 5.63T + 73T^{2} \) |
| 79 | \( 1 - 5.97T + 79T^{2} \) |
| 83 | \( 1 - 0.557T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 6.28T + 97T^{2} \) |
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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
−8.472626105952437194608775476334, −7.76690049695602002719805036069, −6.48318909285918563715825032802, −6.35203361813539977129489452674, −5.59835583023537300803162899317, −5.01983825224772335025695089863, −4.27973266231458582450845737823, −2.34869822242303852920244104820, −1.54295101467236910290473750331, −0.999421903911826603561999094852,
0.999421903911826603561999094852, 1.54295101467236910290473750331, 2.34869822242303852920244104820, 4.27973266231458582450845737823, 5.01983825224772335025695089863, 5.59835583023537300803162899317, 6.35203361813539977129489452674, 6.48318909285918563715825032802, 7.76690049695602002719805036069, 8.472626105952437194608775476334
