L(s) = 1 | + 2-s + 3.37·3-s + 4-s − 0.592·5-s + 3.37·6-s + 0.944·7-s + 8-s + 8.37·9-s − 0.592·10-s + 1.34·11-s + 3.37·12-s − 3.87·13-s + 0.944·14-s − 1.99·15-s + 16-s + 0.703·17-s + 8.37·18-s + 6.97·19-s − 0.592·20-s + 3.18·21-s + 1.34·22-s − 1.28·23-s + 3.37·24-s − 4.64·25-s − 3.87·26-s + 18.1·27-s + 0.944·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.94·3-s + 0.5·4-s − 0.265·5-s + 1.37·6-s + 0.357·7-s + 0.353·8-s + 2.79·9-s − 0.187·10-s + 0.406·11-s + 0.973·12-s − 1.07·13-s + 0.252·14-s − 0.516·15-s + 0.250·16-s + 0.170·17-s + 1.97·18-s + 1.60·19-s − 0.132·20-s + 0.695·21-s + 0.287·22-s − 0.267·23-s + 0.688·24-s − 0.929·25-s − 0.759·26-s + 3.48·27-s + 0.178·28-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
\(L(1)\) |
\(\approx\) |
\(6.578359810\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.578359810\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.37T + 3T^{2} \) |
| 5 | \( 1 + 0.592T + 5T^{2} \) |
| 7 | \( 1 - 0.944T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 0.703T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 5.93T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 - 7.33T + 73T^{2} \) |
| 79 | \( 1 - 2.51T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 1.81T + 97T^{2} \) |
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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
−8.190898857449193781779567013604, −7.79680540751761240108486646174, −7.21647060919627077791146093374, −6.43679187429461900249320808116, −5.07188039856665514483930324829, −4.56835717946564942083607564126, −3.56268194475105438229503186411, −3.16903057228344103974168550157, −2.22308247155116172509349327168, −1.41790440054737917160011534339,
1.41790440054737917160011534339, 2.22308247155116172509349327168, 3.16903057228344103974168550157, 3.56268194475105438229503186411, 4.56835717946564942083607564126, 5.07188039856665514483930324829, 6.43679187429461900249320808116, 7.21647060919627077791146093374, 7.79680540751761240108486646174, 8.190898857449193781779567013604