| L(s) = 1 | − 1.56e3i·3-s − 3.49e4i·5-s + (−8.16e5 + 1.07e5i)7-s + 2.33e6·9-s + 6.84e6·11-s + 2.90e6i·13-s − 5.46e7·15-s + 6.39e7i·17-s − 7.05e8i·19-s + (1.68e8 + 1.27e9i)21-s + 6.52e8·23-s − 1.22e9·25-s − 1.11e10i·27-s − 1.19e9·29-s + 4.78e10i·31-s + ⋯ |
| L(s) = 1 | − 0.715i·3-s − 0.447i·5-s + (−0.991 + 0.131i)7-s + 0.487·9-s + 0.351·11-s + 0.0463i·13-s − 0.320·15-s + 0.155i·17-s − 0.789i·19-s + (0.0938 + 0.709i)21-s + 0.191·23-s − 0.199·25-s − 1.06i·27-s − 0.0693·29-s + 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.990429453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990429453\) |
| \(L(8)\) |
|
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 | \( 1 + (8.16e5 - 1.07e5i)T \) |
| good | 3 | \( 1 + 1.56e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 - 6.84e6T + 3.79e14T^{2} \) |
| 13 | \( 1 - 2.90e6iT - 3.93e15T^{2} \) |
| 17 | \( 1 - 6.39e7iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 7.05e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 6.52e8T + 1.15e19T^{2} \) |
| 29 | \( 1 + 1.19e9T + 2.97e20T^{2} \) |
| 31 | \( 1 - 4.78e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 6.50e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 2.65e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 1.87e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 1.12e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 - 2.13e12T + 1.37e24T^{2} \) |
| 59 | \( 1 + 1.93e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 7.97e11iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 9.53e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 4.55e12T + 8.27e25T^{2} \) |
| 73 | \( 1 - 9.49e12iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 5.62e12T + 3.68e26T^{2} \) |
| 83 | \( 1 - 1.08e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 1.47e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 - 9.21e11iT - 6.52e27T^{2} \) |
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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
−10.18780234641268087702906546970, −9.304341093840897265560941784250, −8.288887716578364854301485849601, −7.02125065850540348238379476388, −6.44586565409349466147744022881, −5.10508508803603003666543586847, −3.86275926724019172781994059880, −2.64201988343810442021105642049, −1.41558543964173474614653812508, −0.53007817410416619865403980785,
0.75026521571689905835735024080, 2.25688634852995555896994149189, 3.52288449125146893321452092375, 4.16124053632925357935148908334, 5.59762268341187121029002482640, 6.63668957731817810477695090518, 7.60383669105760576709789195637, 9.078124540593519100402745335950, 9.877891920510599879363006599681, 10.52364972753044039000792195291
