| L(s) = 1 | − 4·7-s − 6·11-s + 4·13-s + 3·17-s + 7·19-s − 9·23-s + 7·31-s − 2·37-s + 6·41-s + 2·43-s + 9·49-s − 9·53-s + 12·59-s − 7·61-s + 2·67-s − 6·71-s − 2·73-s + 24·77-s + 79-s + 9·83-s − 6·89-s − 16·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s + 1.10·13-s + 0.727·17-s + 1.60·19-s − 1.87·23-s + 1.25·31-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 9/7·49-s − 1.23·53-s + 1.56·59-s − 0.896·61-s + 0.244·67-s − 0.712·71-s − 0.234·73-s + 2.73·77-s + 0.112·79-s + 0.987·83-s − 0.635·89-s − 1.67·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.42645009483319, −16.13031464226039, −15.81520760832704, −15.48305529025095, −14.39275667427376, −13.74438842853409, −13.45893036791924, −12.85302043640720, −12.26172091653426, −11.75223692504549, −10.86124328655187, −10.29781475650687, −9.822555875843269, −9.414309136783613, −8.384837184423652, −7.920883464688669, −7.348546517162633, −6.468975085046080, −5.834871328087318, −5.494963307356017, −4.465277661304055, −3.523552537955942, −3.095791337576131, −2.344820953337232, −1.031848989300437, 0,
1.031848989300437, 2.344820953337232, 3.095791337576131, 3.523552537955942, 4.465277661304055, 5.494963307356017, 5.834871328087318, 6.468975085046080, 7.348546517162633, 7.920883464688669, 8.384837184423652, 9.414309136783613, 9.822555875843269, 10.29781475650687, 10.86124328655187, 11.75223692504549, 12.26172091653426, 12.85302043640720, 13.45893036791924, 13.74438842853409, 14.39275667427376, 15.48305529025095, 15.81520760832704, 16.13031464226039, 16.42645009483319
