| L(s) = 1 | − 2-s − 0.498·3-s + 4-s + 0.121·5-s + 0.498·6-s − 4.42·7-s − 8-s − 2.75·9-s − 0.121·10-s − 2.32·11-s − 0.498·12-s − 1.79·13-s + 4.42·14-s − 0.0604·15-s + 16-s + 0.739·17-s + 2.75·18-s − 7.11·19-s + 0.121·20-s + 2.20·21-s + 2.32·22-s − 1.00·23-s + 0.498·24-s − 4.98·25-s + 1.79·26-s + 2.86·27-s − 4.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.287·3-s + 0.5·4-s + 0.0542·5-s + 0.203·6-s − 1.67·7-s − 0.353·8-s − 0.917·9-s − 0.0383·10-s − 0.701·11-s − 0.143·12-s − 0.498·13-s + 1.18·14-s − 0.0156·15-s + 0.250·16-s + 0.179·17-s + 0.648·18-s − 1.63·19-s + 0.0271·20-s + 0.481·21-s + 0.496·22-s − 0.209·23-s + 0.101·24-s − 0.997·25-s + 0.352·26-s + 0.551·27-s − 0.836·28-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
| \(L(1)\) |
\(\approx\) |
\(0.09418319004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09418319004\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.498T + 3T^{2} \) |
| 5 | \( 1 - 0.121T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 + 1.79T + 13T^{2} \) |
| 17 | \( 1 - 0.739T + 17T^{2} \) |
| 19 | \( 1 + 7.11T + 19T^{2} \) |
| 23 | \( 1 + 1.00T + 23T^{2} \) |
| 29 | \( 1 + 5.80T + 29T^{2} \) |
| 31 | \( 1 - 2.80T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 + 0.415T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 0.00603T + 53T^{2} \) |
| 59 | \( 1 + 0.658T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 2.10T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 - 9.03T + 97T^{2} \) |
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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
−8.483273721844188292291212403969, −7.81255977600753677460655065398, −6.94039065762379044982823318558, −6.17981472532254964549225805159, −5.88278816533227428381818776308, −4.76608085937007438905823546904, −3.57473364022836574815093633335, −2.85889206002659119574605397382, −2.02460639065612269536266254497, −0.18093422058317903818008402844,
0.18093422058317903818008402844, 2.02460639065612269536266254497, 2.85889206002659119574605397382, 3.57473364022836574815093633335, 4.76608085937007438905823546904, 5.88278816533227428381818776308, 6.17981472532254964549225805159, 6.94039065762379044982823318558, 7.81255977600753677460655065398, 8.483273721844188292291212403969
