| L(s) = 1 | − 2.74i·3-s + 1.20i·5-s − 3.93i·7-s − 4.53·9-s − 1.10i·11-s − 0.0851·13-s + 3.30·15-s + (3.92 − 1.25i)17-s + 7.38·19-s − 10.8·21-s + 5.67i·23-s + 3.55·25-s + 4.21i·27-s − 5.59i·29-s − 5.38i·31-s + ⋯ |
| L(s) = 1 | − 1.58i·3-s + 0.537i·5-s − 1.48i·7-s − 1.51·9-s − 0.333i·11-s − 0.0236·13-s + 0.852·15-s + (0.952 − 0.304i)17-s + 1.69·19-s − 2.35·21-s + 1.18i·23-s + 0.710·25-s + 0.810i·27-s − 1.03i·29-s − 0.967i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.938700455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938700455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-3.92 + 1.25i)T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.74iT - 3T^{2} \) |
| 5 | \( 1 - 1.20iT - 5T^{2} \) |
| 7 | \( 1 + 3.93iT - 7T^{2} \) |
| 11 | \( 1 + 1.10iT - 11T^{2} \) |
| 13 | \( 1 + 0.0851T + 13T^{2} \) |
| 19 | \( 1 - 7.38T + 19T^{2} \) |
| 23 | \( 1 - 5.67iT - 23T^{2} \) |
| 29 | \( 1 + 5.59iT - 29T^{2} \) |
| 31 | \( 1 + 5.38iT - 31T^{2} \) |
| 37 | \( 1 + 7.96iT - 37T^{2} \) |
| 41 | \( 1 + 7.91iT - 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 - 3.82T + 53T^{2} \) |
| 61 | \( 1 + 2.00iT - 61T^{2} \) |
| 67 | \( 1 + 8.91T + 67T^{2} \) |
| 71 | \( 1 - 8.59iT - 71T^{2} \) |
| 73 | \( 1 - 7.38iT - 73T^{2} \) |
| 79 | \( 1 - 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 15.6iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62774936622977192072518754606, −7.31627086533610321909771686954, −7.12035634497900233421046930738, −5.93706549667870383847562118485, −5.50546535485963749130552173475, −4.09326508401528238038797693404, −3.32973207474663907735496339641, −2.44072081471783622073281966713, −1.21920014560868358473894996785, −0.65672289987713038109273314674,
1.34652419433164178455093540493, 2.87833401575047337069929559834, 3.22347285788880977348253192684, 4.46344406112291486020534365434, 5.03785251800281797640878463491, 5.46534741082180544532695007837, 6.28917390688059246741202982644, 7.44910371300335462220169450535, 8.472045674805936488268251696504, 8.849224313013710411328751746263
