| L(s) = 1 | + 1.16·2-s − 0.638·4-s − 1.66·5-s − 0.970·7-s − 3.07·8-s − 1.94·10-s + 3.13·11-s + 3.36·13-s − 1.13·14-s − 2.31·16-s − 3.46·17-s + 3.42·19-s + 1.06·20-s + 3.65·22-s + 23-s − 2.23·25-s + 3.92·26-s + 0.619·28-s + 29-s − 6.67·31-s + 3.45·32-s − 4.04·34-s + 1.61·35-s − 3.48·37-s + 3.99·38-s + 5.12·40-s + 7.11·41-s + ⋯ |
| L(s) = 1 | + 0.825·2-s − 0.319·4-s − 0.744·5-s − 0.366·7-s − 1.08·8-s − 0.614·10-s + 0.943·11-s + 0.933·13-s − 0.302·14-s − 0.578·16-s − 0.840·17-s + 0.785·19-s + 0.237·20-s + 0.778·22-s + 0.208·23-s − 0.446·25-s + 0.770·26-s + 0.117·28-s + 0.185·29-s − 1.19·31-s + 0.610·32-s − 0.693·34-s + 0.273·35-s − 0.572·37-s + 0.648·38-s + 0.810·40-s + 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 0.970T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 37 | \( 1 + 3.48T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 - 6.42T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 1.90T + 61T^{2} \) |
| 67 | \( 1 + 0.345T + 67T^{2} \) |
| 71 | \( 1 + 9.30T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 3.43T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + 8.02T + 97T^{2} \) |
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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
−7.64130089738963484201874461486, −6.90570028535916511187710430986, −6.13465031993285335739048943940, −5.60979681524552037028774797067, −4.63340272022332008654253224231, −3.93891406249969218150124667164, −3.61378910585857994573400198823, −2.66609552357521578210166966653, −1.25913244539314970333823178733, 0,
1.25913244539314970333823178733, 2.66609552357521578210166966653, 3.61378910585857994573400198823, 3.93891406249969218150124667164, 4.63340272022332008654253224231, 5.60979681524552037028774797067, 6.13465031993285335739048943940, 6.90570028535916511187710430986, 7.64130089738963484201874461486
