| L(s) = 1 | − i·3-s + (−0.953 − 2.02i)5-s + 5.11i·7-s − 9-s + 3.39·11-s + 5.34i·13-s + (−2.02 + 0.953i)15-s + 1.17i·17-s + 4.64·19-s + 5.11·21-s − 8.93i·23-s + (−3.17 + 3.85i)25-s + i·27-s − 7.07·29-s − 9.13·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.426 − 0.904i)5-s + 1.93i·7-s − 0.333·9-s + 1.02·11-s + 1.48i·13-s + (−0.522 + 0.246i)15-s + 0.284i·17-s + 1.06·19-s + 1.11·21-s − 1.86i·23-s + (−0.635 + 0.771i)25-s + 0.192i·27-s − 1.31·29-s − 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8942795338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8942795338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.953 + 2.02i)T \) |
| 67 | \( 1 + iT \) |
| good | 7 | \( 1 - 5.11iT - 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 - 5.34iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 - 4.64T + 19T^{2} \) |
| 23 | \( 1 + 8.93iT - 23T^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 + 9.13T + 31T^{2} \) |
| 37 | \( 1 - 5.98iT - 37T^{2} \) |
| 41 | \( 1 - 6.63T + 41T^{2} \) |
| 43 | \( 1 - 7.21iT - 43T^{2} \) |
| 47 | \( 1 - 3.89iT - 47T^{2} \) |
| 53 | \( 1 + 5.25iT - 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 0.377T + 61T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 0.211iT - 73T^{2} \) |
| 79 | \( 1 + 0.440T + 79T^{2} \) |
| 83 | \( 1 - 1.58iT - 83T^{2} \) |
| 89 | \( 1 - 7.78T + 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
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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
−8.827134667992159219550878658008, −8.088056512365733970109954053418, −7.24030077362879268951604048210, −6.32236502047386680838842526051, −5.86137348894524034689578251829, −4.95700366787589508926044762463, −4.24004774496795746199303830253, −3.15483248718756557746787018821, −2.05831674961837051763138483239, −1.42526615681896335652131050040,
0.26059526607886294606792086047, 1.45338630056634354206845248445, 3.09108409935776163624173420471, 3.74706555020959915314578118512, 3.95679621198802477585191041413, 5.25912068728688964309819990041, 5.91284016783729231852692162200, 7.07936500116283595897480353955, 7.48243791219186313882914171224, 7.79375236056003969528785455721
