| L(s) = 1 | − 0.101i·3-s + 2.19i·5-s − 7-s + 2.98·9-s − 0.674i·11-s + 5.30i·13-s + 0.222·15-s + 3.87·17-s + 0.995i·19-s + 0.101i·21-s + 3.36·23-s + 0.175·25-s − 0.607i·27-s − 0.134i·29-s − 3.11·31-s + ⋯ |
| L(s) = 1 | − 0.0585i·3-s + 0.982i·5-s − 0.377·7-s + 0.996·9-s − 0.203i·11-s + 1.47i·13-s + 0.0574·15-s + 0.939·17-s + 0.228i·19-s + 0.0221i·21-s + 0.701·23-s + 0.0351·25-s − 0.116i·27-s − 0.0249i·29-s − 0.559·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.893295051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.893295051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.101iT - 3T^{2} \) |
| 5 | \( 1 - 2.19iT - 5T^{2} \) |
| 11 | \( 1 + 0.674iT - 11T^{2} \) |
| 13 | \( 1 - 5.30iT - 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 0.995iT - 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 + 0.134iT - 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 - 2.07iT - 37T^{2} \) |
| 41 | \( 1 - 3.76T + 41T^{2} \) |
| 43 | \( 1 + 3.61iT - 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 - 2.93iT - 53T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 + 4.30iT - 61T^{2} \) |
| 67 | \( 1 - 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 0.856T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.794541687869906419756865215472, −7.85493585219910923038091769075, −6.95830101494218836179919380027, −6.83604058857913857353154063224, −5.90379701296110586411904225851, −4.89811690119107694246444174715, −3.97394385203201640691217269122, −3.30820958567950096904092821385, −2.29009030251839758496519447548, −1.24578223069331265721692351006,
0.63428714638189666668426307039, 1.50211160741313947387627796284, 2.88052399944982813162410583870, 3.69250201001427171398171275634, 4.68479098769623146576430877930, 5.23751597923660699044060043669, 6.01144872942244227203390380064, 7.02304771566758577961083459382, 7.68189832272164405549734110725, 8.317296974450532726331169553193
