| L(s) = 1 | + 1.98·2-s − 3-s + 1.92·4-s − 0.252·5-s − 1.98·6-s − 1.21·7-s − 0.138·8-s + 9-s − 0.501·10-s + 2.73·11-s − 1.92·12-s − 6.01·13-s − 2.40·14-s + 0.252·15-s − 4.13·16-s + 0.539·17-s + 1.98·18-s + 3.87·19-s − 0.488·20-s + 1.21·21-s + 5.43·22-s − 23-s + 0.138·24-s − 4.93·25-s − 11.9·26-s − 27-s − 2.34·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.964·4-s − 0.113·5-s − 0.809·6-s − 0.459·7-s − 0.0491·8-s + 0.333·9-s − 0.158·10-s + 0.826·11-s − 0.557·12-s − 1.66·13-s − 0.643·14-s + 0.0653·15-s − 1.03·16-s + 0.130·17-s + 0.467·18-s + 0.889·19-s − 0.109·20-s + 0.265·21-s + 1.15·22-s − 0.208·23-s + 0.0283·24-s − 0.987·25-s − 2.33·26-s − 0.192·27-s − 0.443·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 0.252T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 - 0.539T + 17T^{2} \) |
| 19 | \( 1 - 3.87T + 19T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 4.48T + 37T^{2} \) |
| 41 | \( 1 + 7.22T + 41T^{2} \) |
| 43 | \( 1 - 6.16T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 4.78T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 0.281T + 73T^{2} \) |
| 79 | \( 1 + 9.98T + 79T^{2} \) |
| 83 | \( 1 + 0.778T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.997494892677023639476334247242, −7.44135370482547600985287008970, −7.11173507754223012787697643340, −6.05780900262705546362320370768, −5.50886298403738272776788316687, −4.72911918494117168612359789348, −3.91531688450865480979864471666, −3.13500161215015654243094005280, −1.93394106997571707737905869840, 0,
1.93394106997571707737905869840, 3.13500161215015654243094005280, 3.91531688450865480979864471666, 4.72911918494117168612359789348, 5.50886298403738272776788316687, 6.05780900262705546362320370768, 7.11173507754223012787697643340, 7.44135370482547600985287008970, 8.997494892677023639476334247242
