| L(s) = 1 | + 0.194·2-s + 3-s − 1.96·4-s + 4.10·5-s + 0.194·6-s + 4.22·7-s − 0.771·8-s + 9-s + 0.799·10-s − 3.86·11-s − 1.96·12-s − 2.41·13-s + 0.822·14-s + 4.10·15-s + 3.77·16-s − 0.0649·17-s + 0.194·18-s + 5.23·19-s − 8.05·20-s + 4.22·21-s − 0.751·22-s + 23-s − 0.771·24-s + 11.8·25-s − 0.470·26-s + 27-s − 8.28·28-s + ⋯ |
| L(s) = 1 | + 0.137·2-s + 0.577·3-s − 0.981·4-s + 1.83·5-s + 0.0794·6-s + 1.59·7-s − 0.272·8-s + 0.333·9-s + 0.252·10-s − 1.16·11-s − 0.566·12-s − 0.669·13-s + 0.219·14-s + 1.05·15-s + 0.943·16-s − 0.0157·17-s + 0.0458·18-s + 1.20·19-s − 1.80·20-s + 0.921·21-s − 0.160·22-s + 0.208·23-s − 0.157·24-s + 2.37·25-s − 0.0922·26-s + 0.192·27-s − 1.56·28-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
| \(L(1)\) |
\(\approx\) |
\(2.994011763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994011763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.194T + 2T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 0.0649T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 31 | \( 1 - 0.653T + 31T^{2} \) |
| 37 | \( 1 - 4.96T + 37T^{2} \) |
| 41 | \( 1 + 1.84T + 41T^{2} \) |
| 43 | \( 1 + 8.33T + 43T^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 + 6.80T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 + 0.532T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−9.293700130091840118795459968269, −8.295395755021847717100386014024, −7.959029592991791859207902636904, −6.82831379685140117350014939198, −5.50786765763813615178477859639, −5.19537148984417405175575400584, −4.59621767555388065136590534153, −3.08752841924676548649410876750, −2.19495904945251971564654882817, −1.25413669415418901006054657701,
1.25413669415418901006054657701, 2.19495904945251971564654882817, 3.08752841924676548649410876750, 4.59621767555388065136590534153, 5.19537148984417405175575400584, 5.50786765763813615178477859639, 6.82831379685140117350014939198, 7.959029592991791859207902636904, 8.295395755021847717100386014024, 9.293700130091840118795459968269
