| L(s) = 1 | + (0.477 + 0.218i)2-s + (0.663 − 2.25i)3-s + (−1.12 − 1.30i)4-s + (−2.15 + 0.587i)5-s + (0.810 − 0.934i)6-s + (3.03 + 0.436i)7-s + (−0.551 − 1.87i)8-s + (−2.13 − 1.37i)9-s + (−1.15 − 0.190i)10-s + (0.639 + 1.39i)11-s + (−3.69 + 1.68i)12-s + (2.46 − 0.354i)13-s + (1.35 + 0.872i)14-s + (−0.103 + 5.26i)15-s + (−0.344 + 2.39i)16-s + (−0.765 − 0.663i)17-s + ⋯ |
| L(s) = 1 | + (0.337 + 0.154i)2-s + (0.382 − 1.30i)3-s + (−0.564 − 0.651i)4-s + (−0.964 + 0.262i)5-s + (0.330 − 0.381i)6-s + (1.14 + 0.165i)7-s + (−0.194 − 0.663i)8-s + (−0.713 − 0.458i)9-s + (−0.366 − 0.0600i)10-s + (0.192 + 0.422i)11-s + (−1.06 + 0.486i)12-s + (0.683 − 0.0983i)13-s + (0.362 + 0.233i)14-s + (−0.0266 + 1.35i)15-s + (−0.0860 + 0.598i)16-s + (−0.185 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.960394 - 0.674280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960394 - 0.674280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.15 - 0.587i)T \) |
| 23 | \( 1 + (4.70 - 0.944i)T \) |
| good | 2 | \( 1 + (-0.477 - 0.218i)T + (1.30 + 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.663 + 2.25i)T + (-2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (-3.03 - 0.436i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.639 - 1.39i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-2.46 + 0.354i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (0.765 + 0.663i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.34 - 5.01i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.91 + 2.21i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (8.97 - 2.63i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (4.50 - 7.00i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (2.93 - 1.88i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 10.3i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 6.01iT - 47T^{2} \) |
| 53 | \( 1 + (0.472 + 0.0679i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.15 - 8.04i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.28 + 0.670i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-4.81 - 2.19i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (3.69 - 8.08i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.01 + 2.61i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.08 + 7.51i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (8.17 - 12.7i)T + (-34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-7.12 - 2.09i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.20 - 3.43i)T + (-40.2 + 88.2i)T^{2} \) |
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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
−13.60179380809760405555479680690, −12.36444239391445021839980794315, −11.66503798325663719239610051114, −10.31768898141983876773435159626, −8.660842945148317471037127374125, −7.86405507004326673618804170334, −6.84113337838523425899605735011, −5.39297006766533233979604448156, −3.88202590685957596195482783401, −1.54510706305523341684486917924,
3.42115172629379583462579837645, 4.26323946046550720015455801820, 5.12447136128338256046350320858, 7.61937667939790814035185472100, 8.567164075133527405027813180016, 9.267206690525943372905854607082, 10.96723905037648422275174932641, 11.46693472649119604641644956955, 12.73450750439258834442630849345, 14.03285334747424561169038508536
