| L(s) = 1 | + (2.70e3 + 4.67e3i)5-s + (4.00e3 − 4.42e4i)7-s + (3.40e5 + 1.96e5i)11-s + 2.36e6i·13-s + (−3.94e6 + 6.83e6i)17-s + (−5.92e6 + 3.42e6i)19-s + (−2.50e7 + 1.44e7i)23-s + (9.82e6 − 1.70e7i)25-s − 4.75e7i·29-s + (−1.45e8 − 8.38e7i)31-s + (2.17e8 − 1.00e8i)35-s + (−2.06e8 − 3.58e8i)37-s + 1.32e9·41-s − 1.71e8·43-s + (−3.33e7 − 5.78e7i)47-s + ⋯ |
| L(s) = 1 | + (0.386 + 0.669i)5-s + (0.0900 − 0.995i)7-s + (0.637 + 0.368i)11-s + 1.76i·13-s + (−0.673 + 1.16i)17-s + (−0.548 + 0.316i)19-s + (−0.811 + 0.468i)23-s + (0.201 − 0.348i)25-s − 0.430i·29-s + (−0.911 − 0.526i)31-s + (0.701 − 0.324i)35-s + (−0.490 − 0.850i)37-s + 1.78·41-s − 0.178·43-s + (−0.0212 − 0.0367i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.2522169225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2522169225\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-4.00e3 + 4.42e4i)T \) |
| good | 5 | \( 1 + (-2.70e3 - 4.67e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-3.40e5 - 1.96e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 2.36e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (3.94e6 - 6.83e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (5.92e6 - 3.42e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (2.50e7 - 1.44e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 4.75e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (1.45e8 + 8.38e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (2.06e8 + 3.58e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.32e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.71e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (3.33e7 + 5.78e7i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (6.71e8 + 3.87e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (2.25e9 - 3.90e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (5.99e8 - 3.46e8i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-2.47e8 + 4.28e8i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.34e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (1.37e10 + 7.95e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-1.08e10 - 1.87e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 2.48e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (1.70e10 + 2.95e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 7.99e9iT - 7.15e21T^{2} \) |
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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
−10.67694605283440151865712025566, −9.763995941558007446192413770958, −8.871903531094541304506815617564, −7.57895647688636778849896393303, −6.70635464226140474991534341432, −6.06251277783226615563794039221, −4.25944493778123313332810450292, −3.93699841240432219166764972241, −2.19810696491340857850371185886, −1.52728481741575707959426128924,
0.04500855904587068209483883544, 1.07669616365874963281362631306, 2.29277171945736386170338308914, 3.26707587582856418420103403776, 4.77982764420827161232923288500, 5.50815536970109867508452761652, 6.42643347999158315404999792331, 7.79002030624382539672628997245, 8.792645557603663338557733170877, 9.293366665058184133349292552012
