| L(s) = 1 | − 2.30·2-s + 2.53·3-s + 3.30·4-s + 3.98·5-s − 5.84·6-s − 4.26·7-s − 3.01·8-s + 3.44·9-s − 9.17·10-s − 11-s + 8.39·12-s + 0.239·13-s + 9.82·14-s + 10.1·15-s + 0.324·16-s + 1.10·17-s − 7.92·18-s + 1.67·19-s + 13.1·20-s − 10.8·21-s + 2.30·22-s − 3.87·23-s − 7.64·24-s + 10.8·25-s − 0.552·26-s + 1.12·27-s − 14.1·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 1.46·3-s + 1.65·4-s + 1.78·5-s − 2.38·6-s − 1.61·7-s − 1.06·8-s + 1.14·9-s − 2.90·10-s − 0.301·11-s + 2.42·12-s + 0.0664·13-s + 2.62·14-s + 2.60·15-s + 0.0810·16-s + 0.268·17-s − 1.86·18-s + 0.383·19-s + 2.94·20-s − 2.36·21-s + 0.491·22-s − 0.808·23-s − 1.56·24-s + 2.17·25-s − 0.108·26-s + 0.215·27-s − 2.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.844116716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.844116716\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 3.98T + 5T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 13 | \( 1 - 0.239T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 + 6.26T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 + 7.48T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 0.664T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.574715639518743386952552154988, −7.39496472370840991882160350654, −7.07412238558046318256798484310, −6.09856477991213307947153589143, −5.71233118389524793596083838776, −4.15084001756334166608539400875, −2.98908389746692542933871300606, −2.58658719615393805986659469680, −1.91520780939064802795744212671, −0.844647061797011678109137313571,
0.844647061797011678109137313571, 1.91520780939064802795744212671, 2.58658719615393805986659469680, 2.98908389746692542933871300606, 4.15084001756334166608539400875, 5.71233118389524793596083838776, 6.09856477991213307947153589143, 7.07412238558046318256798484310, 7.39496472370840991882160350654, 8.574715639518743386952552154988
