| L(s) = 1 | + (−2.83 − 0.978i)3-s + (4.69 − 2.71i)7-s + (7.08 + 5.54i)9-s + (−8.81 + 5.09i)11-s + (4.42 + 2.55i)13-s − 17.4·17-s + 17.4·19-s + (−15.9 + 3.09i)21-s + (9.41 − 16.3i)23-s + (−14.6 − 22.6i)27-s + (−29.0 + 16.7i)29-s + (−25.4 + 44.1i)31-s + (29.9 − 5.81i)33-s + 0.605i·37-s + (−10.0 − 11.5i)39-s + ⋯ |
| L(s) = 1 | + (−0.945 − 0.326i)3-s + (0.671 − 0.387i)7-s + (0.787 + 0.616i)9-s + (−0.801 + 0.462i)11-s + (0.340 + 0.196i)13-s − 1.02·17-s + 0.920·19-s + (−0.760 + 0.147i)21-s + (0.409 − 0.709i)23-s + (−0.543 − 0.839i)27-s + (−1.00 + 0.578i)29-s + (−0.822 + 1.42i)31-s + (0.908 − 0.176i)33-s + 0.0163i·37-s + (−0.257 − 0.296i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9999242025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9999242025\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.83 + 0.978i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-4.69 + 2.71i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.81 - 5.09i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-4.42 - 2.55i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 17.4T + 289T^{2} \) |
| 19 | \( 1 - 17.4T + 361T^{2} \) |
| 23 | \( 1 + (-9.41 + 16.3i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (29.0 - 16.7i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (25.4 - 44.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 0.605iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (12.4 + 7.17i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-44.0 + 25.4i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.13 + 1.96i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-70.8 - 40.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.70 - 8.14i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-31.8 - 18.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 57.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.1 - 33.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-38.6 - 66.9i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 138. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (101. - 58.4i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.40320758768238648415966127809, −9.260303807599972632979911577206, −8.267461927688969013473805346840, −7.26966481172083117119399055939, −6.84247189683856057814196822483, −5.52754389902536672979107670337, −4.97171618086732584323265497533, −3.95844517501598709422920905416, −2.29477121945595149521245434456, −1.10588676795560288333538262549,
0.42459812086619264104170905878, 1.96260548126661093097166207436, 3.46305424639011278626723704278, 4.59013262915210176457063501350, 5.45606870886931763082307816976, 6.00093414862671658664062083457, 7.22592542633089053596132075231, 7.988116259921828809019144973703, 9.083870364102018377734516461547, 9.781624263940454113131443044138
