| L(s) = 1 | − 1.24·2-s − 0.198·3-s − 0.445·4-s − 5-s + 0.246·6-s + 0.198·7-s + 3.04·8-s − 2.96·9-s + 1.24·10-s + 4.04·11-s + 0.0881·12-s − 0.246·14-s + 0.198·15-s − 2.91·16-s − 5.40·17-s + 3.69·18-s + 2.18·19-s + 0.445·20-s − 0.0392·21-s − 5.04·22-s + 7.23·23-s − 0.603·24-s + 25-s + 1.18·27-s − 0.0881·28-s − 7.07·29-s − 0.246·30-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 0.114·3-s − 0.222·4-s − 0.447·5-s + 0.100·6-s + 0.0748·7-s + 1.07·8-s − 0.986·9-s + 0.394·10-s + 1.22·11-s + 0.0254·12-s − 0.0660·14-s + 0.0511·15-s − 0.727·16-s − 1.31·17-s + 0.870·18-s + 0.501·19-s + 0.0995·20-s − 0.00856·21-s − 1.07·22-s + 1.50·23-s − 0.123·24-s + 0.200·25-s + 0.227·27-s − 0.0166·28-s − 1.31·29-s − 0.0450·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 0.198T + 3T^{2} \) |
| 7 | \( 1 - 0.198T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 7.23T + 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 0.0217T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 + 0.554T + 53T^{2} \) |
| 59 | \( 1 - 3.89T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 9.52T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 - 5.02T + 97T^{2} \) |
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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
−9.475990990148765310195829248210, −8.921104542057642890393484580090, −8.385507185324287798357653916288, −7.30496045329322420049594607665, −6.56383853678374264262425755661, −5.26473872947049779101897030095, −4.34989189445464241463462383712, −3.22501672459389089202879873345, −1.53849933836578933275795459049, 0,
1.53849933836578933275795459049, 3.22501672459389089202879873345, 4.34989189445464241463462383712, 5.26473872947049779101897030095, 6.56383853678374264262425755661, 7.30496045329322420049594607665, 8.385507185324287798357653916288, 8.921104542057642890393484580090, 9.475990990148765310195829248210
