| L(s) = 1 | − 2.02·2-s + 2.08·4-s + 3.61·5-s + 7-s − 0.164·8-s − 7.30·10-s − 1.71·11-s + 0.128·13-s − 2.02·14-s − 3.83·16-s − 6.97·17-s + 2.84·19-s + 7.52·20-s + 3.46·22-s − 7.59·23-s + 8.06·25-s − 0.259·26-s + 2.08·28-s − 2.66·29-s − 6.90·31-s + 8.06·32-s + 14.0·34-s + 3.61·35-s + 3.46·37-s − 5.74·38-s − 0.592·40-s + 3.74·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.04·4-s + 1.61·5-s + 0.377·7-s − 0.0579·8-s − 2.30·10-s − 0.517·11-s + 0.0356·13-s − 0.539·14-s − 0.957·16-s − 1.69·17-s + 0.652·19-s + 1.68·20-s + 0.738·22-s − 1.58·23-s + 1.61·25-s − 0.0509·26-s + 0.393·28-s − 0.494·29-s − 1.24·31-s + 1.42·32-s + 2.41·34-s + 0.611·35-s + 0.568·37-s − 0.932·38-s − 0.0937·40-s + 0.585·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 - 0.128T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 3.74T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 7.71T + 53T^{2} \) |
| 59 | \( 1 - 2.24T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 6.26T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 8.03T + 79T^{2} \) |
| 83 | \( 1 + 1.88T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.59534306314336378445388148168, −6.96123455098914694923630948021, −6.24076977945527483118664073819, −5.56946101811322202140405567953, −4.85627524673671764721765852005, −3.89293036889680418639994038998, −2.34461527668030628749051478673, −2.17799817854800761943509192346, −1.27308794877082216581891051395, 0,
1.27308794877082216581891051395, 2.17799817854800761943509192346, 2.34461527668030628749051478673, 3.89293036889680418639994038998, 4.85627524673671764721765852005, 5.56946101811322202140405567953, 6.24076977945527483118664073819, 6.96123455098914694923630948021, 7.59534306314336378445388148168
