| L(s) = 1 | + 2.53i·3-s + 1.47i·5-s − 7-s − 3.43·9-s + 5.35i·11-s + 3.55i·13-s − 3.73·15-s − 2.73·17-s − 8.61i·19-s − 2.53i·21-s − 6.17·23-s + 2.83·25-s − 1.09i·27-s − 8.49i·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | + 1.46i·3-s + 0.658i·5-s − 0.377·7-s − 1.14·9-s + 1.61i·11-s + 0.985i·13-s − 0.963·15-s − 0.664·17-s − 1.97i·19-s − 0.553i·21-s − 1.28·23-s + 0.567·25-s − 0.210i·27-s − 1.57i·29-s − 1.79·31-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
| \(L(1)\) |
\(\approx\) |
\(0.5076687563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5076687563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.53iT - 3T^{2} \) |
| 5 | \( 1 - 1.47iT - 5T^{2} \) |
| 11 | \( 1 - 5.35iT - 11T^{2} \) |
| 13 | \( 1 - 3.55iT - 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 + 8.61iT - 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 8.49iT - 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 0.333iT - 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 - 5.82iT - 43T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 - 1.20iT - 53T^{2} \) |
| 59 | \( 1 + 2.71iT - 59T^{2} \) |
| 61 | \( 1 - 4.48iT - 61T^{2} \) |
| 67 | \( 1 - 8.45iT - 67T^{2} \) |
| 71 | \( 1 + 2.46T + 71T^{2} \) |
| 73 | \( 1 - 6.31T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 5.41iT - 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−9.293711107807162258214381214238, −8.741147471327939646383956919886, −7.38455810926672435159787217406, −6.98591968080555144154864688186, −6.12920309272995441044458135111, −5.09776510459784151974686994662, −4.26278230673211200201348970235, −4.09941270586556302857561600195, −2.78668210291082141935781760732, −2.10639684052027061878830740915,
0.15511514666221401260724208090, 1.14276419225469808023194467824, 2.01634241358481846613041751986, 3.19926564132450371930927420432, 3.87813388538474784035311552700, 5.40003236634047324015169800528, 5.79550904178169685531883479772, 6.45501965161370404901230874844, 7.38345267371080970966429588950, 7.979817749719561964193865693361
