| L(s) = 1 | + 2·11-s + 17-s − 4·19-s − 4·23-s − 5·25-s + 8·31-s + 4·37-s + 6·41-s + 8·43-s − 8·47-s − 7·49-s − 10·53-s + 12·61-s − 8·67-s + 12·71-s − 2·73-s − 4·79-s + 16·83-s + 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.603·11-s + 0.242·17-s − 0.917·19-s − 0.834·23-s − 25-s + 1.43·31-s + 0.657·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 49-s − 1.37·53-s + 1.53·61-s − 0.977·67-s + 1.42·71-s − 0.234·73-s − 0.450·79-s + 1.75·83-s + 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326320065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326320065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.88069481378399, −12.71348996922867, −11.95606768106480, −11.72594593896857, −11.20234523874328, −10.72628165676250, −10.15431631301065, −9.689863068000118, −9.406412370807990, −8.744091838173916, −8.225814360602580, −7.829787036400547, −7.446045755459543, −6.557357625004260, −6.298835552592513, −5.983058281151245, −5.209886948982841, −4.585032463326311, −4.237470081797724, −3.638722246593626, −3.105281915826999, −2.290078115498909, −1.956116262683502, −1.114512932150557, −0.4605549077446753,
0.4605549077446753, 1.114512932150557, 1.956116262683502, 2.290078115498909, 3.105281915826999, 3.638722246593626, 4.237470081797724, 4.585032463326311, 5.209886948982841, 5.983058281151245, 6.298835552592513, 6.557357625004260, 7.446045755459543, 7.829787036400547, 8.225814360602580, 8.744091838173916, 9.406412370807990, 9.689863068000118, 10.15431631301065, 10.72628165676250, 11.20234523874328, 11.72594593896857, 11.95606768106480, 12.71348996922867, 12.88069481378399
