| L(s) = 1 | + (1.35 − 0.784i)3-s + (−42.3 + 24.5i)7-s + (−39.2 + 68.0i)9-s + (−11.7 − 20.3i)11-s + 136. i·13-s + (227. − 131. i)17-s + (−387. − 223. i)19-s + (−38.3 + 66.6i)21-s + (−374. + 648. i)23-s + 250. i·27-s + 406.·29-s + (−584. + 337. i)31-s + (−31.9 − 18.4i)33-s + (372. − 645. i)37-s + (106. + 185. i)39-s + ⋯ |
| L(s) = 1 | + (0.151 − 0.0871i)3-s + (−0.865 + 0.501i)7-s + (−0.484 + 0.839i)9-s + (−0.0971 − 0.168i)11-s + 0.806i·13-s + (0.786 − 0.454i)17-s + (−1.07 − 0.619i)19-s + (−0.0869 + 0.151i)21-s + (−0.708 + 1.22i)23-s + 0.343i·27-s + 0.483·29-s + (−0.607 + 0.350i)31-s + (−0.0293 − 0.0169i)33-s + (0.272 − 0.471i)37-s + (0.0702 + 0.121i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6308628824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6308628824\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (42.3 - 24.5i)T \) |
| good | 3 | \( 1 + (-1.35 + 0.784i)T + (40.5 - 70.1i)T^{2} \) |
| 11 | \( 1 + (11.7 + 20.3i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 136. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-227. + 131. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (387. + 223. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (374. - 648. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 406.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (584. - 337. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-372. + 645. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.47e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.63e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (-579. - 334. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (1.01e3 + 1.75e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.01e3 - 585. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.02e3 + 1.16e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (3.77e3 + 6.54e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.57e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (3.28e3 - 1.89e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.89e3 - 6.75e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 2.18e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-8.05e3 - 4.64e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 9.98e3iT - 8.85e7T^{2} \) |
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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
−9.459296262633537291954463061776, −8.908966996313616291990643782074, −7.898084714242182738215987624151, −7.02581090703651825341119300307, −6.00158330215999678819000323134, −5.23629723408520391011011178030, −3.96476017225166711647526941868, −2.84677377488613590682003206551, −1.92340010697878974088420952080, −0.17080237525447218596066879154,
0.925147972311194636441000135732, 2.60618712788548907243407062803, 3.53653544930177053112784322258, 4.41285007244037864396454336851, 5.96657097696311834476979089516, 6.31218761245838807087655850024, 7.57873765781313149783827202116, 8.366096559867994967872433292203, 9.295551205739028599675098390533, 10.19347325513605957120612172365
