| L(s) = 1 | + 1.15·2-s − 0.658·4-s − 2.87·5-s + 3.24·7-s − 3.07·8-s − 3.33·10-s + 2.26·11-s − 1.37·13-s + 3.76·14-s − 2.24·16-s − 8.07·17-s − 1.03·19-s + 1.89·20-s + 2.62·22-s − 6.96·23-s + 3.28·25-s − 1.58·26-s − 2.14·28-s − 3.91·29-s − 3.54·31-s + 3.55·32-s − 9.34·34-s − 9.35·35-s − 2.26·37-s − 1.19·38-s + 8.86·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | + 0.818·2-s − 0.329·4-s − 1.28·5-s + 1.22·7-s − 1.08·8-s − 1.05·10-s + 0.682·11-s − 0.380·13-s + 1.00·14-s − 0.562·16-s − 1.95·17-s − 0.237·19-s + 0.424·20-s + 0.559·22-s − 1.45·23-s + 0.657·25-s − 0.311·26-s − 0.404·28-s − 0.727·29-s − 0.635·31-s + 0.628·32-s − 1.60·34-s − 1.58·35-s − 0.371·37-s − 0.194·38-s + 1.40·40-s + 1.72·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 + 8.07T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 9.25T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 + 3.10T + 53T^{2} \) |
| 59 | \( 1 + 3.63T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 + 2.50T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 1.48T + 79T^{2} \) |
| 83 | \( 1 - 5.76T + 83T^{2} \) |
| 89 | \( 1 - 9.29T + 89T^{2} \) |
| 97 | \( 1 + 9.32T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.495095825248087339933311740246, −8.640636921661152799982138449001, −8.063558163807597578204652903104, −7.09810517338898079330658024294, −6.05152847710391132664964132510, −4.86176409188083612744683218965, −4.30144944654255330456463621669, −3.67753287534649215740943397242, −2.08721872626007956365082063973, 0,
2.08721872626007956365082063973, 3.67753287534649215740943397242, 4.30144944654255330456463621669, 4.86176409188083612744683218965, 6.05152847710391132664964132510, 7.09810517338898079330658024294, 8.063558163807597578204652903104, 8.640636921661152799982138449001, 9.495095825248087339933311740246
