| L(s) = 1 | + 0.0852·2-s + 0.824·3-s − 1.99·4-s − 5-s + 0.0702·6-s + 0.193·7-s − 0.340·8-s − 2.32·9-s − 0.0852·10-s − 11-s − 1.64·12-s + 2.33·13-s + 0.0164·14-s − 0.824·15-s + 3.95·16-s + 0.0234·17-s − 0.197·18-s + 2.93·19-s + 1.99·20-s + 0.159·21-s − 0.0852·22-s − 0.350·23-s − 0.280·24-s + 25-s + 0.199·26-s − 4.38·27-s − 0.385·28-s + ⋯ |
| L(s) = 1 | + 0.0602·2-s + 0.475·3-s − 0.996·4-s − 0.447·5-s + 0.0286·6-s + 0.0730·7-s − 0.120·8-s − 0.773·9-s − 0.0269·10-s − 0.301·11-s − 0.474·12-s + 0.648·13-s + 0.00440·14-s − 0.212·15-s + 0.989·16-s + 0.00569·17-s − 0.0466·18-s + 0.673·19-s + 0.445·20-s + 0.0347·21-s − 0.0181·22-s − 0.0730·23-s − 0.0572·24-s + 0.200·25-s + 0.0391·26-s − 0.844·27-s − 0.0728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0852T + 2T^{2} \) |
| 3 | \( 1 - 0.824T + 3T^{2} \) |
| 7 | \( 1 - 0.193T + 7T^{2} \) |
| 13 | \( 1 - 2.33T + 13T^{2} \) |
| 17 | \( 1 - 0.0234T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 0.350T + 23T^{2} \) |
| 29 | \( 1 - 2.05T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + 3.17T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.291474176026800044741458014580, −7.67970137872840430013449565452, −6.55917290543659694936644563297, −5.74254956643796467382888101160, −4.99845294014693331387197397194, −4.23444947282196282908741921791, −3.38369570389027588396288397985, −2.78277848448194052299966582079, −1.27765748884258850395291194723, 0,
1.27765748884258850395291194723, 2.78277848448194052299966582079, 3.38369570389027588396288397985, 4.23444947282196282908741921791, 4.99845294014693331387197397194, 5.74254956643796467382888101160, 6.55917290543659694936644563297, 7.67970137872840430013449565452, 8.291474176026800044741458014580
