| L(s) = 1 | + 1.17·2-s − 0.630·4-s − 0.290·7-s − 3.07·8-s + 0.460·11-s + 1.80·13-s − 0.340·14-s − 2.34·16-s − 3.09·17-s + 3·19-s + 0.539·22-s − 2.46·23-s + 2.10·26-s + 0.183·28-s + 9.58·29-s + 31-s + 3.41·32-s − 3.61·34-s − 6.87·37-s + 3.51·38-s − 3.32·41-s − 10.8·43-s − 0.290·44-s − 2.87·46-s + 10.3·47-s − 6.91·49-s − 1.13·52-s + ⋯ |
| L(s) = 1 | + 0.827·2-s − 0.315·4-s − 0.109·7-s − 1.08·8-s + 0.138·11-s + 0.499·13-s − 0.0909·14-s − 0.585·16-s − 0.749·17-s + 0.688·19-s + 0.114·22-s − 0.513·23-s + 0.413·26-s + 0.0346·28-s + 1.78·29-s + 0.179·31-s + 0.604·32-s − 0.620·34-s − 1.13·37-s + 0.569·38-s − 0.519·41-s − 1.65·43-s − 0.0438·44-s − 0.424·46-s + 1.50·47-s − 0.987·49-s − 0.157·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 7 | \( 1 + 0.290T + 7T^{2} \) |
| 11 | \( 1 - 0.460T + 11T^{2} \) |
| 13 | \( 1 - 1.80T + 13T^{2} \) |
| 17 | \( 1 + 3.09T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 2.46T + 23T^{2} \) |
| 29 | \( 1 - 9.58T + 29T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 + 1.92T + 53T^{2} \) |
| 59 | \( 1 - 0.986T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 0.0289T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 9.91T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 5.53T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−7.54688567788475905713659969812, −6.53990033046792016285008051359, −6.28015763317819638869172978197, −5.25964019809501516870195114975, −4.80976486230793698535133709427, −3.98670498258579528766396980018, −3.34506204885810724904478227889, −2.56218003353601339035917740097, −1.32135640832396013377385105055, 0,
1.32135640832396013377385105055, 2.56218003353601339035917740097, 3.34506204885810724904478227889, 3.98670498258579528766396980018, 4.80976486230793698535133709427, 5.25964019809501516870195114975, 6.28015763317819638869172978197, 6.53990033046792016285008051359, 7.54688567788475905713659969812
