| L(s) = 1 | + 2.44·2-s + 3-s + 3.97·4-s − 1.74·5-s + 2.44·6-s + 5.10·7-s + 4.84·8-s + 9-s − 4.26·10-s − 5.64·11-s + 3.97·12-s + 6.72·13-s + 12.4·14-s − 1.74·15-s + 3.87·16-s − 0.427·17-s + 2.44·18-s − 1.22·19-s − 6.94·20-s + 5.10·21-s − 13.7·22-s + 23-s + 4.84·24-s − 1.95·25-s + 16.4·26-s + 27-s + 20.3·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.577·3-s + 1.98·4-s − 0.780·5-s + 0.998·6-s + 1.92·7-s + 1.71·8-s + 0.333·9-s − 1.34·10-s − 1.70·11-s + 1.14·12-s + 1.86·13-s + 3.33·14-s − 0.450·15-s + 0.969·16-s − 0.103·17-s + 0.576·18-s − 0.281·19-s − 1.55·20-s + 1.11·21-s − 2.94·22-s + 0.208·23-s + 0.987·24-s − 0.390·25-s + 3.22·26-s + 0.192·27-s + 3.83·28-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
| \(L(1)\) |
\(\approx\) |
\(6.299302528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.299302528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 - 5.10T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 + 0.427T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 + 4.88T + 43T^{2} \) |
| 47 | \( 1 + 8.92T + 47T^{2} \) |
| 53 | \( 1 - 5.99T + 53T^{2} \) |
| 59 | \( 1 - 0.440T + 59T^{2} \) |
| 61 | \( 1 + 6.08T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 - 3.56T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.12T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 + 8.18T + 97T^{2} \) |
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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
−8.688300026256511557532711994649, −8.140301978657640452456253838261, −7.69578943074126808509653300764, −6.67254302993883752122842172120, −5.57998256519970821983669413345, −4.98561952220859320124882691310, −4.25910104136078221229361591144, −3.55369107941093317031895213141, −2.55616702367193452609209588715, −1.56758925097933240637642518541,
1.56758925097933240637642518541, 2.55616702367193452609209588715, 3.55369107941093317031895213141, 4.25910104136078221229361591144, 4.98561952220859320124882691310, 5.57998256519970821983669413345, 6.67254302993883752122842172120, 7.69578943074126808509653300764, 8.140301978657640452456253838261, 8.688300026256511557532711994649
