| L(s) = 1 | − 3.89e3i·3-s − 3.49e4i·5-s + (−8.06e5 + 1.68e5i)7-s − 1.03e7·9-s − 1.36e7·11-s + 3.11e7i·13-s − 1.35e8·15-s + 1.27e8i·17-s + 1.25e9i·19-s + (6.56e8 + 3.13e9i)21-s − 1.28e9·23-s − 1.22e9·25-s + 2.17e10i·27-s − 1.34e10·29-s − 4.30e9i·31-s + ⋯ |
| L(s) = 1 | − 1.77i·3-s − 0.447i·5-s + (−0.978 + 0.204i)7-s − 2.16·9-s − 0.700·11-s + 0.496i·13-s − 0.795·15-s + 0.309i·17-s + 1.40i·19-s + (0.364 + 1.74i)21-s − 0.376·23-s − 0.199·25-s + 2.07i·27-s − 0.779·29-s − 0.156i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.9179771031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9179771031\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (8.06e5 - 1.68e5i)T \) |
| good | 3 | \( 1 + 3.89e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 1.36e7T + 3.79e14T^{2} \) |
| 13 | \( 1 - 3.11e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 - 1.27e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.25e9iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 1.28e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 1.34e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 4.30e9iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 3.62e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + 1.83e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 2.10e10T + 7.38e22T^{2} \) |
| 47 | \( 1 + 9.67e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 7.91e11T + 1.37e24T^{2} \) |
| 59 | \( 1 - 2.02e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 3.80e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 6.56e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 2.60e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + 7.69e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 3.17e13T + 3.68e26T^{2} \) |
| 83 | \( 1 + 3.51e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 2.09e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 2.50e13iT - 6.52e27T^{2} \) |
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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
−10.36466219454534788009493294633, −9.072929788382946702129095584121, −8.093966107308733121538666562448, −7.27175007922312780700678191083, −6.24016909569159325950846783364, −5.54590065015754798578661197887, −3.68546399473996753279582924316, −2.40289721893471238525137128274, −1.60540174681537378894460043202, −0.44136624190539667970037225149,
0.35866833665439437852390201502, 2.69394013614845344356405234198, 3.27332523630283988935624235056, 4.38095181677642996812421637852, 5.32926882506859261881321588034, 6.42608506118199631191431157161, 7.84561672510863858103072389283, 9.229830101766591537206570274579, 9.750132879014308826881111744829, 10.68680906365533619711203755995
