| L(s) = 1 | + 3-s + 3.93·5-s + 2.32·7-s + 9-s − 1.68·11-s − 13-s + 3.93·15-s + 4.13·17-s − 0.414·19-s + 2.32·21-s + 3.75·23-s + 10.5·25-s + 27-s − 8.12·29-s + 10.7·31-s − 1.68·33-s + 9.17·35-s + 3.43·37-s − 39-s + 4.95·41-s + 7.15·43-s + 3.93·45-s − 3.53·47-s − 1.58·49-s + 4.13·51-s − 9.70·53-s − 6.62·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.76·5-s + 0.879·7-s + 0.333·9-s − 0.506·11-s − 0.277·13-s + 1.01·15-s + 1.00·17-s − 0.0951·19-s + 0.508·21-s + 0.783·23-s + 2.10·25-s + 0.192·27-s − 1.50·29-s + 1.92·31-s − 0.292·33-s + 1.55·35-s + 0.564·37-s − 0.160·39-s + 0.773·41-s + 1.09·43-s + 0.587·45-s − 0.515·47-s − 0.225·49-s + 0.579·51-s − 1.33·53-s − 0.892·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.689073747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.689073747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 - 2.32T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 17 | \( 1 - 4.13T + 17T^{2} \) |
| 19 | \( 1 + 0.414T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 + 8.12T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 - 4.95T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 + 9.70T + 53T^{2} \) |
| 59 | \( 1 + 3.77T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 8.63T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 9.09T + 97T^{2} \) |
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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
−7.82463848810598282572776007018, −6.99473052211994094074823814877, −6.19277163553944696834040463518, −5.58764710775397396078316356937, −5.00783619086664837333170397858, −4.33896561779675189408537296754, −3.08731569466443526702432627811, −2.55916954347899054549495616320, −1.76278304248348247021936533770, −1.08955433031255600799000882242,
1.08955433031255600799000882242, 1.76278304248348247021936533770, 2.55916954347899054549495616320, 3.08731569466443526702432627811, 4.33896561779675189408537296754, 5.00783619086664837333170397858, 5.58764710775397396078316356937, 6.19277163553944696834040463518, 6.99473052211994094074823814877, 7.82463848810598282572776007018
