| L(s) = 1 | + 2.26·2-s + 3-s + 3.12·4-s + 0.128·5-s + 2.26·6-s − 4.08·7-s + 2.54·8-s + 9-s + 0.291·10-s + 5.61·11-s + 3.12·12-s + 0.0956·13-s − 9.25·14-s + 0.128·15-s − 0.487·16-s − 0.474·17-s + 2.26·18-s − 0.856·19-s + 0.401·20-s − 4.08·21-s + 12.7·22-s − 7.09·23-s + 2.54·24-s − 4.98·25-s + 0.216·26-s + 27-s − 12.7·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.56·4-s + 0.0575·5-s + 0.924·6-s − 1.54·7-s + 0.899·8-s + 0.333·9-s + 0.0920·10-s + 1.69·11-s + 0.901·12-s + 0.0265·13-s − 2.47·14-s + 0.0332·15-s − 0.121·16-s − 0.115·17-s + 0.533·18-s − 0.196·19-s + 0.0898·20-s − 0.892·21-s + 2.70·22-s − 1.48·23-s + 0.519·24-s − 0.996·25-s + 0.0424·26-s + 0.192·27-s − 2.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175277337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175277337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 5 | \( 1 - 0.128T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 - 0.0956T + 13T^{2} \) |
| 17 | \( 1 + 0.474T + 17T^{2} \) |
| 19 | \( 1 + 0.856T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 8.11T + 29T^{2} \) |
| 31 | \( 1 - 1.18T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 - 7.00T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 8.43T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 0.225T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 3.47T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 + 0.891T + 89T^{2} \) |
| 97 | \( 1 - 1.89T + 97T^{2} \) |
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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
−12.19417783492411400973483910487, −11.10601080396503027238459436795, −9.661035425888528503681356129204, −9.145600044786956960421655305312, −7.46588874861544502894590096886, −6.38048025993227925905309298594, −5.90395841096139971291941971381, −4.03733170385255415052832118868, −3.73184007156455556823388973218, −2.33347917820590274013982739812,
2.33347917820590274013982739812, 3.73184007156455556823388973218, 4.03733170385255415052832118868, 5.90395841096139971291941971381, 6.38048025993227925905309298594, 7.46588874861544502894590096886, 9.145600044786956960421655305312, 9.661035425888528503681356129204, 11.10601080396503027238459436795, 12.19417783492411400973483910487
