L(s) = 1 | + 0.946·2-s − 1.10·4-s + 4.23·5-s + 2.19·7-s − 2.93·8-s + 4.00·10-s − 2.38·11-s − 5.96·13-s + 2.07·14-s − 0.570·16-s + 4.21·17-s + 5.07·19-s − 4.67·20-s − 2.25·22-s + 0.949·23-s + 12.9·25-s − 5.64·26-s − 2.42·28-s + 0.816·29-s + 9.04·31-s + 5.33·32-s + 3.99·34-s + 9.29·35-s + 9.83·37-s + 4.80·38-s − 12.4·40-s + 4.51·41-s + ⋯ |
L(s) = 1 | + 0.669·2-s − 0.552·4-s + 1.89·5-s + 0.829·7-s − 1.03·8-s + 1.26·10-s − 0.717·11-s − 1.65·13-s + 0.555·14-s − 0.142·16-s + 1.02·17-s + 1.16·19-s − 1.04·20-s − 0.480·22-s + 0.197·23-s + 2.58·25-s − 1.10·26-s − 0.458·28-s + 0.151·29-s + 1.62·31-s + 0.943·32-s + 0.684·34-s + 1.57·35-s + 1.61·37-s + 0.779·38-s − 1.96·40-s + 0.704·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.084590006\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.084590006\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
good | 2 | \( 1 - 0.946T + 2T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 - 0.949T + 23T^{2} \) |
| 29 | \( 1 - 0.816T + 29T^{2} \) |
| 31 | \( 1 - 9.04T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 2.93T + 61T^{2} \) |
| 67 | \( 1 - 9.58T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 7.39T + 73T^{2} \) |
| 79 | \( 1 + 8.43T + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 3.33T + 97T^{2} \) |
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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
−9.374224722667276259357958076773, −8.288481914131073261277109424216, −7.57065548480157639332843101899, −6.42479830784893832777491292786, −5.63464551050471377807840334734, −5.03825677026832100307037406151, −4.70253515189088944614633888172, −3.02281134511018921301406258528, −2.44101788409633765202696333194, −1.12771464448261886233080464413,
1.12771464448261886233080464413, 2.44101788409633765202696333194, 3.02281134511018921301406258528, 4.70253515189088944614633888172, 5.03825677026832100307037406151, 5.63464551050471377807840334734, 6.42479830784893832777491292786, 7.57065548480157639332843101899, 8.288481914131073261277109424216, 9.374224722667276259357958076773