| L(s) = 1 | + 1.36·3-s − 2.50·5-s − 7-s − 1.13·9-s + 11-s + 13-s − 3.42·15-s − 6.95·17-s − 2.13·19-s − 1.36·21-s + 3.66·23-s + 1.28·25-s − 5.64·27-s + 7.07·29-s + 3.87·31-s + 1.36·33-s + 2.50·35-s + 1.84·37-s + 1.36·39-s + 2.04·41-s + 3.91·43-s + 2.84·45-s − 7.44·47-s + 49-s − 9.48·51-s − 3.73·53-s − 2.50·55-s + ⋯ |
| L(s) = 1 | + 0.788·3-s − 1.12·5-s − 0.377·7-s − 0.378·9-s + 0.301·11-s + 0.277·13-s − 0.883·15-s − 1.68·17-s − 0.490·19-s − 0.297·21-s + 0.763·23-s + 0.256·25-s − 1.08·27-s + 1.31·29-s + 0.695·31-s + 0.237·33-s + 0.423·35-s + 0.303·37-s + 0.218·39-s + 0.319·41-s + 0.596·43-s + 0.424·45-s − 1.08·47-s + 0.142·49-s − 1.32·51-s − 0.512·53-s − 0.337·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364967185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364967185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 2.50T + 5T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 + 2.13T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 - 3.91T + 43T^{2} \) |
| 47 | \( 1 + 7.44T + 47T^{2} \) |
| 53 | \( 1 + 3.73T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.48T + 61T^{2} \) |
| 67 | \( 1 - 2.87T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−8.013875296162783697022208916256, −7.21530561643131873115963208626, −6.53865261369381608169627703755, −5.95775661847920638074849165664, −4.60383337247410751835586863283, −4.36618111956721498709497530667, −3.35024414011689741205939525805, −2.89557832291847459700064180225, −1.94248094561944986697166995438, −0.53260817183614215126087350810,
0.53260817183614215126087350810, 1.94248094561944986697166995438, 2.89557832291847459700064180225, 3.35024414011689741205939525805, 4.36618111956721498709497530667, 4.60383337247410751835586863283, 5.95775661847920638074849165664, 6.53865261369381608169627703755, 7.21530561643131873115963208626, 8.013875296162783697022208916256
