| L(s) = 1 | + 0.754·2-s + 1.25·3-s − 1.43·4-s + 5-s + 0.944·6-s − 7-s − 2.58·8-s − 1.43·9-s + 0.754·10-s + 0.402·11-s − 1.78·12-s + 0.640·13-s − 0.754·14-s + 1.25·15-s + 0.907·16-s − 0.0100·17-s − 1.08·18-s + 3.82·19-s − 1.43·20-s − 1.25·21-s + 0.303·22-s − 6.75·23-s − 3.23·24-s + 25-s + 0.483·26-s − 5.54·27-s + 1.43·28-s + ⋯ |
| L(s) = 1 | + 0.533·2-s + 0.722·3-s − 0.715·4-s + 0.447·5-s + 0.385·6-s − 0.377·7-s − 0.915·8-s − 0.478·9-s + 0.238·10-s + 0.121·11-s − 0.516·12-s + 0.177·13-s − 0.201·14-s + 0.323·15-s + 0.226·16-s − 0.00244·17-s − 0.255·18-s + 0.877·19-s − 0.319·20-s − 0.273·21-s + 0.0647·22-s − 1.40·23-s − 0.661·24-s + 0.200·25-s + 0.0947·26-s − 1.06·27-s + 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.754T + 2T^{2} \) |
| 3 | \( 1 - 1.25T + 3T^{2} \) |
| 11 | \( 1 - 0.402T + 11T^{2} \) |
| 13 | \( 1 - 0.640T + 13T^{2} \) |
| 17 | \( 1 + 0.0100T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 9.03T + 29T^{2} \) |
| 31 | \( 1 + 5.27T + 31T^{2} \) |
| 37 | \( 1 - 7.77T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 - 5.21T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.0689T + 71T^{2} \) |
| 73 | \( 1 + 5.55T + 73T^{2} \) |
| 79 | \( 1 - 0.720T + 79T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 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
−7.76596569048603588407254595806, −6.57620960246847917016207215365, −5.96544524888748180936837358580, −5.46797084792045944226415084413, −4.53249881384800526228143044594, −3.90456450192124763770050154622, −3.06679095744642505511321998925, −2.61631761069846895213710881539, −1.35127807517237949267490865929, 0,
1.35127807517237949267490865929, 2.61631761069846895213710881539, 3.06679095744642505511321998925, 3.90456450192124763770050154622, 4.53249881384800526228143044594, 5.46797084792045944226415084413, 5.96544524888748180936837358580, 6.57620960246847917016207215365, 7.76596569048603588407254595806
