| L(s) = 1 | + 0.603·2-s + 3-s − 1.63·4-s − 2.98·5-s + 0.603·6-s + 4.14·7-s − 2.19·8-s + 9-s − 1.80·10-s − 1.78·11-s − 1.63·12-s − 3.75·13-s + 2.50·14-s − 2.98·15-s + 1.94·16-s + 3.76·17-s + 0.603·18-s + 2.44·19-s + 4.88·20-s + 4.14·21-s − 1.08·22-s + 23-s − 2.19·24-s + 3.90·25-s − 2.26·26-s + 27-s − 6.77·28-s + ⋯ |
| L(s) = 1 | + 0.427·2-s + 0.577·3-s − 0.817·4-s − 1.33·5-s + 0.246·6-s + 1.56·7-s − 0.776·8-s + 0.333·9-s − 0.570·10-s − 0.539·11-s − 0.472·12-s − 1.04·13-s + 0.668·14-s − 0.770·15-s + 0.486·16-s + 0.912·17-s + 0.142·18-s + 0.560·19-s + 1.09·20-s + 0.903·21-s − 0.230·22-s + 0.208·23-s − 0.448·24-s + 0.781·25-s − 0.444·26-s + 0.192·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.817178931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.817178931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.603T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 3.75T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 31 | \( 1 + 0.393T + 31T^{2} \) |
| 37 | \( 1 + 0.656T + 37T^{2} \) |
| 41 | \( 1 + 7.56T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 7.63T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 2.39T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 0.336T + 89T^{2} \) |
| 97 | \( 1 - 8.25T + 97T^{2} \) |
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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
−8.926336846627703079271699091167, −8.247501152034837975443073332483, −7.71406210413055813283301213378, −7.26154910529224986603254285900, −5.53204561757002893612875111243, −4.95836797493066032276573496948, −4.26119980066784253188957218748, −3.53029312338832589936745737640, −2.45450090510757371460357747043, −0.841246478205455246607994161795,
0.841246478205455246607994161795, 2.45450090510757371460357747043, 3.53029312338832589936745737640, 4.26119980066784253188957218748, 4.95836797493066032276573496948, 5.53204561757002893612875111243, 7.26154910529224986603254285900, 7.71406210413055813283301213378, 8.247501152034837975443073332483, 8.926336846627703079271699091167
