| L(s) = 1 | − 1.01i·3-s − 1.29i·5-s − 7-s + 1.96·9-s + 3.99i·11-s + 0.534i·13-s − 1.31·15-s − 6.47·17-s + 3.19i·19-s + 1.01i·21-s − 2.28·23-s + 3.33·25-s − 5.04i·27-s − 6.09i·29-s + 5.70·31-s + ⋯ |
| L(s) = 1 | − 0.587i·3-s − 0.577i·5-s − 0.377·7-s + 0.655·9-s + 1.20i·11-s + 0.148i·13-s − 0.339·15-s − 1.57·17-s + 0.731i·19-s + 0.221i·21-s − 0.476·23-s + 0.666·25-s − 0.971i·27-s − 1.13i·29-s + 1.02·31-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
| \(L(1)\) |
\(\approx\) |
\(1.588332300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588332300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.01iT - 3T^{2} \) |
| 5 | \( 1 + 1.29iT - 5T^{2} \) |
| 11 | \( 1 - 3.99iT - 11T^{2} \) |
| 13 | \( 1 - 0.534iT - 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 3.19iT - 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 + 6.09iT - 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 + 6.44iT - 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 5.11iT - 43T^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 + 13.5iT - 53T^{2} \) |
| 59 | \( 1 + 9.71iT - 59T^{2} \) |
| 61 | \( 1 + 2.06iT - 61T^{2} \) |
| 67 | \( 1 - 9.06iT - 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 8.18T + 73T^{2} \) |
| 79 | \( 1 - 0.960T + 79T^{2} \) |
| 83 | \( 1 + 5.40iT - 83T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−8.300194602081093795232305092624, −7.59292866578484781514728028917, −6.82653125613356079351891297632, −6.39943278009171218347320805134, −5.32328896690118784810135482318, −4.38388394646566743177387951052, −3.99540648508792006356219421627, −2.39623698343811447256982662553, −1.84664226657096333075493802583, −0.54759631864609384479997406617,
1.03113843439868617471462026605, 2.57191353768418209238508536279, 3.18026747092456610472447911661, 4.17795857944093154983126454702, 4.76753176621698700583203447702, 5.82757857546037977176379821651, 6.59358225938469070053976381126, 7.06235550194062422539213752885, 8.086363136681743981777744214533, 8.924469854962468332821334345362
