| L(s) = 1 | + 2.76i·2-s − i·3-s − 5.66·4-s − i·5-s + 2.76·6-s + (2.02 + 1.70i)7-s − 10.1i·8-s − 9-s + 2.76·10-s + (−3.06 − 1.27i)11-s + 5.66i·12-s − 3.06·13-s + (−4.72 + 5.59i)14-s − 15-s + 16.7·16-s + 2.89·17-s + ⋯ |
| L(s) = 1 | + 1.95i·2-s − 0.577i·3-s − 2.83·4-s − 0.447i·5-s + 1.13·6-s + (0.764 + 0.644i)7-s − 3.58i·8-s − 0.333·9-s + 0.875·10-s + (−0.922 − 0.385i)11-s + 1.63i·12-s − 0.850·13-s + (−1.26 + 1.49i)14-s − 0.258·15-s + 4.19·16-s + 0.702·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.300654886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300654886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.02 - 1.70i)T \) |
| 11 | \( 1 + (3.06 + 1.27i)T \) |
| good | 2 | \( 1 - 2.76iT - 2T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 - 3.97iT - 29T^{2} \) |
| 31 | \( 1 - 5.54iT - 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 - 9.03T + 41T^{2} \) |
| 43 | \( 1 + 0.633iT - 43T^{2} \) |
| 47 | \( 1 + 2.57iT - 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 6.56iT - 59T^{2} \) |
| 61 | \( 1 - 3.39T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 - 5.94iT - 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 17.6iT - 89T^{2} \) |
| 97 | \( 1 - 3.11iT - 97T^{2} \) |
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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
−9.500689672929133066603051223820, −8.888287651024684465276872380921, −8.136424125045764713953199168385, −7.55162568683147631975015978921, −6.96311519591970408667050809587, −5.64430028393377449806532364825, −5.37447974402574785317433175992, −4.62862265963527190183955845717, −3.03605947189411751339112966805, −0.968539229745927609881022274954,
0.816657539135524808743227099798, 2.29528314818878241012492998302, 3.05235138999531715157400058524, 4.07409772362832261094745138381, 4.86451590768586511954298267858, 5.49215866270458687659164454033, 7.52602774299685585843202962593, 8.015102666850403178590392899175, 9.321985507469182433195987395492, 9.755000897213052395780116605098
