| L(s) = 1 | + 1.01·3-s − 0.303·5-s − 7-s − 1.97·9-s − 0.882·11-s − 5.59·13-s − 0.306·15-s + 1.54·17-s − 0.0998·19-s − 1.01·21-s − 8.23·23-s − 4.90·25-s − 5.03·27-s + 1.69·29-s + 4.52·31-s − 0.892·33-s + 0.303·35-s + 0.883·37-s − 5.66·39-s + 9.92·41-s + 3.30·43-s + 0.600·45-s − 6.00·47-s + 49-s + 1.56·51-s + 11.1·53-s + 0.267·55-s + ⋯ |
| L(s) = 1 | + 0.583·3-s − 0.135·5-s − 0.377·7-s − 0.659·9-s − 0.266·11-s − 1.55·13-s − 0.0791·15-s + 0.375·17-s − 0.0229·19-s − 0.220·21-s − 1.71·23-s − 0.981·25-s − 0.968·27-s + 0.315·29-s + 0.813·31-s − 0.155·33-s + 0.0512·35-s + 0.145·37-s − 0.906·39-s + 1.55·41-s + 0.503·43-s + 0.0894·45-s − 0.876·47-s + 0.142·49-s + 0.219·51-s + 1.53·53-s + 0.0361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359573630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359573630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 + 0.303T + 5T^{2} \) |
| 11 | \( 1 + 0.882T + 11T^{2} \) |
| 13 | \( 1 + 5.59T + 13T^{2} \) |
| 17 | \( 1 - 1.54T + 17T^{2} \) |
| 19 | \( 1 + 0.0998T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 1.69T + 29T^{2} \) |
| 31 | \( 1 - 4.52T + 31T^{2} \) |
| 37 | \( 1 - 0.883T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 6.00T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.91T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 - 5.05T + 97T^{2} \) |
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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
−7.984029329975506650218036777967, −7.43919157883684892746698473875, −6.54333902906316842798191042302, −5.79803535421020202672017002811, −5.16847537633458651309258055862, −4.18982996606448609308844035445, −3.55953571132409705519854136977, −2.49537380218168180373527428876, −2.24783165949766779098618632230, −0.53494417918196861261425418814,
0.53494417918196861261425418814, 2.24783165949766779098618632230, 2.49537380218168180373527428876, 3.55953571132409705519854136977, 4.18982996606448609308844035445, 5.16847537633458651309258055862, 5.79803535421020202672017002811, 6.54333902906316842798191042302, 7.43919157883684892746698473875, 7.984029329975506650218036777967
