| L(s) = 1 | − 1.30·5-s − 4.60·7-s + 1.30·11-s − 2.30·13-s + 6·17-s − 2·19-s − 6.90·23-s − 3.30·25-s − 6.90·29-s − 3.30·31-s + 6·35-s + 37-s + 0.908·41-s + 6.60·43-s − 2.60·47-s + 14.2·49-s + 6·53-s − 1.69·55-s + 3.39·59-s − 10.5·61-s + 3·65-s − 14.5·67-s + 6·71-s − 8.69·73-s − 6·77-s + 16.1·79-s + 17.2·83-s + ⋯ |
| L(s) = 1 | − 0.582·5-s − 1.74·7-s + 0.392·11-s − 0.638·13-s + 1.45·17-s − 0.458·19-s − 1.44·23-s − 0.660·25-s − 1.28·29-s − 0.593·31-s + 1.01·35-s + 0.164·37-s + 0.141·41-s + 1.00·43-s − 0.380·47-s + 2.03·49-s + 0.824·53-s − 0.228·55-s + 0.441·59-s − 1.34·61-s + 0.372·65-s − 1.77·67-s + 0.712·71-s − 1.01·73-s − 0.683·77-s + 1.81·79-s + 1.88·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7414371140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7414371140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 - 6.60T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 3.39T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.917412893735398735986211858177, −7.58242637061055568943259050906, −6.75874815607977243786170772059, −5.99904347861116558143262700295, −5.53629837794986049556700759798, −4.20930440610602166843192013789, −3.70795194043446432463270997150, −3.02653839150152059887048336757, −1.95889160974842474047576033800, −0.44139049440837343800753970317,
0.44139049440837343800753970317, 1.95889160974842474047576033800, 3.02653839150152059887048336757, 3.70795194043446432463270997150, 4.20930440610602166843192013789, 5.53629837794986049556700759798, 5.99904347861116558143262700295, 6.75874815607977243786170772059, 7.58242637061055568943259050906, 7.917412893735398735986211858177
