| L(s) = 1 | + 5-s + 1.64·7-s + 1.64·11-s + 3.64·13-s + 4.64·17-s + 2.64·19-s + 4.29·23-s + 25-s + 5.64·29-s − 2.64·31-s + 1.64·35-s − 5.29·37-s + 0.354·41-s − 6.93·43-s − 5.29·47-s − 4.29·49-s − 3.93·53-s + 1.64·55-s − 3.64·59-s + 7.58·61-s + 3.64·65-s + 6·67-s + 2.35·71-s − 0.937·73-s + 2.70·77-s − 7.35·79-s − 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.622·7-s + 0.496·11-s + 1.01·13-s + 1.12·17-s + 0.606·19-s + 0.894·23-s + 0.200·25-s + 1.04·29-s − 0.475·31-s + 0.278·35-s − 0.869·37-s + 0.0553·41-s − 1.05·43-s − 0.771·47-s − 0.613·49-s − 0.540·53-s + 0.221·55-s − 0.474·59-s + 0.970·61-s + 0.452·65-s + 0.733·67-s + 0.279·71-s − 0.109·73-s + 0.308·77-s − 0.827·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.776236258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776236258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 - 5.64T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 0.354T + 41T^{2} \) |
| 43 | \( 1 + 6.93T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 + 3.64T + 59T^{2} \) |
| 61 | \( 1 - 7.58T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 2.35T + 71T^{2} \) |
| 73 | \( 1 + 0.937T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 - 8.35T + 89T^{2} \) |
| 97 | \( 1 + 0.708T + 97T^{2} \) |
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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
−8.429734640633928128209843633198, −7.71955060410918404013660062146, −6.85371475617558781053826044100, −6.21156123960467324581281104694, −5.34218035339466289379601461297, −4.81822628699013227694833187279, −3.67142048843788867103821109690, −3.05335292036821086574160110597, −1.72776685930917524782033072565, −1.05266372443566364214128136595,
1.05266372443566364214128136595, 1.72776685930917524782033072565, 3.05335292036821086574160110597, 3.67142048843788867103821109690, 4.81822628699013227694833187279, 5.34218035339466289379601461297, 6.21156123960467324581281104694, 6.85371475617558781053826044100, 7.71955060410918404013660062146, 8.429734640633928128209843633198
