| L(s) = 1 | − 2.57·2-s − 3-s + 4.63·4-s + 5-s + 2.57·6-s + 2.22·7-s − 6.79·8-s + 9-s − 2.57·10-s − 3.85·11-s − 4.63·12-s + 5.70·13-s − 5.72·14-s − 15-s + 8.23·16-s − 2.07·17-s − 2.57·18-s + 6.30·19-s + 4.63·20-s − 2.22·21-s + 9.92·22-s + 6.79·24-s + 25-s − 14.7·26-s − 27-s + 10.3·28-s + 7.07·29-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.31·4-s + 0.447·5-s + 1.05·6-s + 0.840·7-s − 2.40·8-s + 0.333·9-s − 0.814·10-s − 1.16·11-s − 1.33·12-s + 1.58·13-s − 1.53·14-s − 0.258·15-s + 2.05·16-s − 0.502·17-s − 0.607·18-s + 1.44·19-s + 1.03·20-s − 0.485·21-s + 2.11·22-s + 1.38·24-s + 0.200·25-s − 2.88·26-s − 0.192·27-s + 1.94·28-s + 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.005582956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005582956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 6.30T + 19T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 - 9.82T + 37T^{2} \) |
| 41 | \( 1 - 9.04T + 41T^{2} \) |
| 43 | \( 1 + 1.69T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.83T + 59T^{2} \) |
| 61 | \( 1 - 6.07T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 - 0.951T + 71T^{2} \) |
| 73 | \( 1 + 7.81T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.978916643738459207821607759859, −7.43991586584526141348135116375, −6.56040493475627236367758433123, −6.03731737825904256829775988723, −5.30428561158758035459255028986, −4.43646823802664140071720264935, −3.07388476022820738993995985191, −2.31618832753276542457532884694, −1.30276615898019949624466147992, −0.78566140822657075057631749279,
0.78566140822657075057631749279, 1.30276615898019949624466147992, 2.31618832753276542457532884694, 3.07388476022820738993995985191, 4.43646823802664140071720264935, 5.30428561158758035459255028986, 6.03731737825904256829775988723, 6.56040493475627236367758433123, 7.43991586584526141348135116375, 7.978916643738459207821607759859
