| L(s) = 1 | − 2-s + 3.03·3-s + 4-s − 2.32·5-s − 3.03·6-s − 0.499·7-s − 8-s + 6.21·9-s + 2.32·10-s − 4.76·11-s + 3.03·12-s + 5.37·13-s + 0.499·14-s − 7.05·15-s + 16-s + 6.96·17-s − 6.21·18-s + 4.21·19-s − 2.32·20-s − 1.51·21-s + 4.76·22-s − 4.83·23-s − 3.03·24-s + 0.397·25-s − 5.37·26-s + 9.76·27-s − 0.499·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.03·5-s − 1.23·6-s − 0.188·7-s − 0.353·8-s + 2.07·9-s + 0.734·10-s − 1.43·11-s + 0.876·12-s + 1.49·13-s + 0.133·14-s − 1.82·15-s + 0.250·16-s + 1.68·17-s − 1.46·18-s + 0.967·19-s − 0.519·20-s − 0.331·21-s + 1.01·22-s − 1.00·23-s − 0.619·24-s + 0.0795·25-s − 1.05·26-s + 1.87·27-s − 0.0944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354951334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354951334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 3.03T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 + 0.499T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 - 0.646T + 29T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 + 8.32T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 0.991T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 2.95T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 5.73T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
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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
−8.234714071853884749251037088853, −7.78977481784763323799079744423, −7.17530792651668626697697326092, −6.11734368248977941909309012488, −5.16932091268777325881239413710, −3.96312188436433715841038273380, −3.40912757607174070626818765428, −2.94720955467235035126652744309, −1.90065910352833079544512493353, −0.838302491289429017508863397947,
0.838302491289429017508863397947, 1.90065910352833079544512493353, 2.94720955467235035126652744309, 3.40912757607174070626818765428, 3.96312188436433715841038273380, 5.16932091268777325881239413710, 6.11734368248977941909309012488, 7.17530792651668626697697326092, 7.78977481784763323799079744423, 8.234714071853884749251037088853
