| L(s) = 1 | − 2.40i·3-s + (−0.817 + 2.08i)5-s − 4.41i·7-s − 2.76·9-s − 2.29·11-s − 6.92i·13-s + (4.99 + 1.96i)15-s + 1.51i·17-s + 2.89·19-s − 10.5·21-s + i·23-s + (−3.66 − 3.40i)25-s − 0.570i·27-s + 7.68·29-s − 3.85·31-s + ⋯ |
| L(s) = 1 | − 1.38i·3-s + (−0.365 + 0.930i)5-s − 1.66i·7-s − 0.920·9-s − 0.691·11-s − 1.92i·13-s + (1.29 + 0.506i)15-s + 0.367i·17-s + 0.665·19-s − 2.31·21-s + 0.208i·23-s + (−0.732 − 0.680i)25-s − 0.109i·27-s + 1.42·29-s − 0.692·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9226702265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9226702265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.817 - 2.08i)T \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.40iT - 3T^{2} \) |
| 7 | \( 1 + 4.41iT - 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 - 1.51iT - 17T^{2} \) |
| 19 | \( 1 - 2.89T + 19T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 8.62iT - 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 3.48iT - 43T^{2} \) |
| 47 | \( 1 - 6.19iT - 47T^{2} \) |
| 53 | \( 1 + 2.17iT - 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 + 9.94iT - 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 8.95iT - 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 8.04iT - 83T^{2} \) |
| 89 | \( 1 - 1.09T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.183592829804519047816183212536, −7.895983718327581382886671335803, −7.29381018324104151361711645050, −6.70147271230336810202012287058, −5.86759281411038218021041246435, −4.70702876780464455578662811212, −3.41231517526350579753469229579, −2.86176984065097590887610397820, −1.36841272747922754528522968270, −0.34816800345945935418330573742,
1.87895918312123885299030802349, 3.01428987617229355193171605935, 4.10408748884012303740693728389, 4.86685181567639933228768271589, 5.29590428032276068253215870361, 6.25450143677515516711832096201, 7.48240281005911424114948788371, 8.570903995265446411353930818643, 9.067432680681139612297630868739, 9.370907306180734342490833234130
