| L(s) = 1 | − 2-s − 2.06·3-s + 4-s + 2.09·5-s + 2.06·6-s + 5.11·7-s − 8-s + 1.27·9-s − 2.09·10-s + 2.69·11-s − 2.06·12-s − 4.18·13-s − 5.11·14-s − 4.33·15-s + 16-s − 1.41·17-s − 1.27·18-s − 1.56·19-s + 2.09·20-s − 10.5·21-s − 2.69·22-s + 2.42·23-s + 2.06·24-s − 0.608·25-s + 4.18·26-s + 3.56·27-s + 5.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.19·3-s + 0.5·4-s + 0.937·5-s + 0.844·6-s + 1.93·7-s − 0.353·8-s + 0.425·9-s − 0.662·10-s + 0.813·11-s − 0.596·12-s − 1.15·13-s − 1.36·14-s − 1.11·15-s + 0.250·16-s − 0.342·17-s − 0.300·18-s − 0.359·19-s + 0.468·20-s − 2.30·21-s − 0.575·22-s + 0.505·23-s + 0.422·24-s − 0.121·25-s + 0.820·26-s + 0.685·27-s + 0.967·28-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
| \(L(1)\) |
\(\approx\) |
\(1.294251989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294251989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.06T + 3T^{2} \) |
| 5 | \( 1 - 2.09T + 5T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 - 2.42T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 0.862T + 47T^{2} \) |
| 53 | \( 1 - 0.339T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2.83T + 61T^{2} \) |
| 67 | \( 1 + 6.55T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.52T + 79T^{2} \) |
| 83 | \( 1 + 3.91T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 1.57T + 97T^{2} \) |
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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
−8.622179494501508580308213813414, −7.49168038531421065258816862005, −7.18443249422127181752260840468, −6.05758682149508206352806987618, −5.60922987157545194090537943471, −4.90480884221790904546857118114, −4.17017134939461671144676529275, −2.40316543667845583379717091782, −1.77997782100875500641340236260, −0.798940921395962186467067490121,
0.798940921395962186467067490121, 1.77997782100875500641340236260, 2.40316543667845583379717091782, 4.17017134939461671144676529275, 4.90480884221790904546857118114, 5.60922987157545194090537943471, 6.05758682149508206352806987618, 7.18443249422127181752260840468, 7.49168038531421065258816862005, 8.622179494501508580308213813414
