L(s) = 1 | + 7-s − 2·11-s − 13-s + 3·17-s + 23-s − 5·29-s − 7·31-s − 2·37-s − 7·41-s + 11·43-s − 8·47-s + 49-s + 53-s + 5·59-s + 3·61-s + 12·67-s + 12·71-s + 6·73-s − 2·77-s − 10·79-s − 11·83-s + 10·89-s − 91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s + 0.208·23-s − 0.928·29-s − 1.25·31-s − 0.328·37-s − 1.09·41-s + 1.67·43-s − 1.16·47-s + 1/7·49-s + 0.137·53-s + 0.650·59-s + 0.384·61-s + 1.46·67-s + 1.42·71-s + 0.702·73-s − 0.227·77-s − 1.12·79-s − 1.20·83-s + 1.05·89-s − 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.21479366605948, −13.34980230171052, −13.04605172956271, −12.57337710812733, −12.07205214136267, −11.49191326675508, −11.03951338447184, −10.64354778053726, −10.00917919381139, −9.589983073214569, −9.086426136502634, −8.420065301469041, −8.023172588438270, −7.459599576440445, −7.069145518805634, −6.439662794183338, −5.669444313691103, −5.296830854448564, −4.932880554079554, −4.029275008686960, −3.637947371884256, −2.939873451674269, −2.235332399222655, −1.707123207879919, −0.8564774153935640, 0,
0.8564774153935640, 1.707123207879919, 2.235332399222655, 2.939873451674269, 3.637947371884256, 4.029275008686960, 4.932880554079554, 5.296830854448564, 5.669444313691103, 6.439662794183338, 7.069145518805634, 7.459599576440445, 8.023172588438270, 8.420065301469041, 9.086426136502634, 9.589983073214569, 10.00917919381139, 10.64354778053726, 11.03951338447184, 11.49191326675508, 12.07205214136267, 12.57337710812733, 13.04605172956271, 13.34980230171052, 14.21479366605948