| L(s) = 1 | + (1.95 − 1.08i)5-s + (−2.37 + 1.16i)7-s − 3.74i·11-s + 0.841·13-s + 3.36i·17-s − 4.55i·19-s − 7.64·23-s + (2.64 − 4.24i)25-s + 1.41i·29-s + 0.979i·31-s + (−3.38 + 4.85i)35-s − 2.32i·37-s − 10.3·41-s + 10.8i·43-s − 7.91i·47-s + ⋯ |
| L(s) = 1 | + (0.874 − 0.485i)5-s + (−0.898 + 0.439i)7-s − 1.12i·11-s + 0.233·13-s + 0.814i·17-s − 1.04i·19-s − 1.59·23-s + (0.529 − 0.848i)25-s + 0.262i·29-s + 0.175i·31-s + (−0.571 + 0.820i)35-s − 0.382i·37-s − 1.61·41-s + 1.64i·43-s − 1.15i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3803783859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3803783859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.95 + 1.08i)T \) |
| 7 | \( 1 + (2.37 - 1.16i)T \) |
| good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 0.841T + 13T^{2} \) |
| 17 | \( 1 - 3.36iT - 17T^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 0.979iT - 31T^{2} \) |
| 37 | \( 1 + 2.32iT - 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 7.91iT - 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 3.14iT - 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−8.210321798303332193343070777231, −6.92478527708475410922373099710, −6.28215737378376192315755299828, −5.82086406496009004049149925004, −5.13454486178219259082853047737, −4.07899556674725288211341107728, −3.23459358788551844062738926334, −2.41441428306146654693909992819, −1.39215037659267202544547208181, −0.094179659458859810424351340034,
1.54965115463739229380535580320, 2.33161551146446426073670650508, 3.29865223514794431649800677659, 4.03907204254973188540502693906, 4.99012958535483099046615004595, 5.87748207854332957890596423209, 6.41723378065849237495118064644, 7.09804117992255190763524414954, 7.70098949571984506007807988777, 8.637010768897551042553834545245
