L(s) = 1 | + 4·7-s − 11-s − 4·13-s + 4·19-s − 6·23-s − 6·29-s + 8·31-s + 2·37-s − 6·41-s + 8·43-s + 6·47-s + 9·49-s + 6·53-s − 12·59-s − 2·61-s − 10·67-s + 12·71-s + 16·73-s − 4·77-s + 8·79-s − 6·89-s − 16·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 0.301·11-s − 1.10·13-s + 0.917·19-s − 1.25·23-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 1.22·67-s + 1.42·71-s + 1.87·73-s − 0.455·77-s + 0.900·79-s − 0.635·89-s − 1.67·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.691199178\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.691199178\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.60789050397075, −12.52285105490286, −12.34380215042433, −11.89468419323373, −11.42202320393599, −10.90415821923476, −10.55738544044131, −9.812135835687902, −9.603368182770631, −8.968606829777659, −8.248255395592855, −7.973906363830217, −7.566076832085738, −7.117254265367193, −6.434426845767496, −5.611743881954056, −5.481793543623491, −4.728183991387453, −4.440610256739815, −3.813891272752462, −3.025581625419178, −2.366415542680781, −1.959706055354715, −1.239785119206941, −0.4920318291269847,
0.4920318291269847, 1.239785119206941, 1.959706055354715, 2.366415542680781, 3.025581625419178, 3.813891272752462, 4.440610256739815, 4.728183991387453, 5.481793543623491, 5.611743881954056, 6.434426845767496, 7.117254265367193, 7.566076832085738, 7.973906363830217, 8.248255395592855, 8.968606829777659, 9.603368182770631, 9.812135835687902, 10.55738544044131, 10.90415821923476, 11.42202320393599, 11.89468419323373, 12.34380215042433, 12.52285105490286, 13.60789050397075