| L(s) = 1 | + 376. i·3-s + 3.49e4i·5-s + (4.89e5 + 6.62e5i)7-s + 4.64e6·9-s − 1.32e7·11-s + 3.65e7i·13-s − 1.31e7·15-s − 4.09e8i·17-s + 8.08e8i·19-s + (−2.49e8 + 1.84e8i)21-s − 2.19e9·23-s − 1.22e9·25-s + 3.54e9i·27-s − 2.49e10·29-s − 1.84e10i·31-s + ⋯ |
| L(s) = 1 | + 0.172i·3-s + 0.447i·5-s + (0.594 + 0.804i)7-s + 0.970·9-s − 0.679·11-s + 0.583i·13-s − 0.0770·15-s − 0.997i·17-s + 0.904i·19-s + (−0.138 + 0.102i)21-s − 0.646·23-s − 0.199·25-s + 0.339i·27-s − 1.44·29-s − 0.671i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.4011026655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4011026655\) |
| \(L(8)\) |
|
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 | \( 1 + (-4.89e5 - 6.62e5i)T \) |
| good | 3 | \( 1 - 376. iT - 4.78e6T^{2} \) |
| 11 | \( 1 + 1.32e7T + 3.79e14T^{2} \) |
| 13 | \( 1 - 3.65e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 4.09e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 8.08e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 + 2.19e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 2.49e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 1.84e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 + 7.52e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 1.36e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 5.14e9T + 7.38e22T^{2} \) |
| 47 | \( 1 - 1.55e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 1.23e12T + 1.37e24T^{2} \) |
| 59 | \( 1 + 3.86e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 - 7.36e10iT - 9.87e24T^{2} \) |
| 67 | \( 1 + 2.57e11T + 3.67e25T^{2} \) |
| 71 | \( 1 + 1.65e13T + 8.27e25T^{2} \) |
| 73 | \( 1 - 7.41e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 1.17e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 1.27e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 9.27e12iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 5.77e13iT - 6.52e27T^{2} \) |
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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
−11.19622032590672131183919110528, −10.10605974677021089295917383860, −9.281841574049087881326297277871, −8.011882844307155873718635573879, −7.17170755581721009825873690325, −5.88372790907457100949058467093, −4.87991439088335677187802757854, −3.76065811599076179160824291183, −2.41167925019984750813942944226, −1.56372326687440701174263605363,
0.07258087019251733495406736678, 1.14382979141033178836365120191, 2.05275795749608786156478774772, 3.66828119210575437459844092581, 4.59573249134898289162998697288, 5.62985495511401964201070122945, 7.08297907121353282239395168334, 7.78790434469945799918696537646, 8.829232661739696483565407860012, 10.19762069926679594928022750574
