| L(s) = 1 | + 1.66·3-s + 3.34·5-s − 0.235·9-s + 3.61·11-s + 2.28·13-s + 5.56·15-s + 4.82·17-s + 1.03·19-s − 23-s + 6.21·25-s − 5.37·27-s − 9.92·29-s − 10.2·31-s + 6.01·33-s + 4.56·37-s + 3.79·39-s + 2.01·41-s + 9.38·43-s − 0.789·45-s + 8.42·47-s + 8.01·51-s − 2.58·53-s + 12.1·55-s + 1.71·57-s + 12.9·59-s − 13.4·61-s + 7.65·65-s + ⋯ |
| L(s) = 1 | + 0.959·3-s + 1.49·5-s − 0.0785·9-s + 1.08·11-s + 0.633·13-s + 1.43·15-s + 1.16·17-s + 0.236·19-s − 0.208·23-s + 1.24·25-s − 1.03·27-s − 1.84·29-s − 1.84·31-s + 1.04·33-s + 0.751·37-s + 0.608·39-s + 0.314·41-s + 1.43·43-s − 0.117·45-s + 1.22·47-s + 1.12·51-s − 0.354·53-s + 1.63·55-s + 0.227·57-s + 1.68·59-s − 1.72·61-s + 0.949·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.651009822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.651009822\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 - 3.34T + 5T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 29 | \( 1 + 9.92T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 9.30T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 7.99T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−7.67127467618502932905216918821, −7.22794986236697848223283214838, −6.06555770069811328578354185077, −5.88471373671299611273516702708, −5.17978129511800537859796191141, −3.80265798522774347221126971054, −3.59419385530489535930561821748, −2.45553612983166995923093878250, −1.90318973529274428905143081076, −1.06326127653659573741659441556,
1.06326127653659573741659441556, 1.90318973529274428905143081076, 2.45553612983166995923093878250, 3.59419385530489535930561821748, 3.80265798522774347221126971054, 5.17978129511800537859796191141, 5.88471373671299611273516702708, 6.06555770069811328578354185077, 7.22794986236697848223283214838, 7.67127467618502932905216918821
