L(s) = 1 | + (0.739 − 1.78i)5-s + (−0.385 + 0.385i)7-s + (−2.36 + 5.70i)11-s + (−2.30 + 0.956i)13-s − 5.05·17-s + (1.27 + 3.07i)19-s + (−2.28 + 2.28i)23-s + (0.892 + 0.892i)25-s + (−0.735 + 0.304i)29-s + 3.40i·31-s + (0.403 + 0.973i)35-s + (9.56 + 3.96i)37-s + (−5.27 − 5.27i)41-s + (−2.53 − 1.05i)43-s − 6.85i·47-s + ⋯ |
L(s) = 1 | + (0.330 − 0.798i)5-s + (−0.145 + 0.145i)7-s + (−0.712 + 1.72i)11-s + (−0.640 + 0.265i)13-s − 1.22·17-s + (0.292 + 0.705i)19-s + (−0.476 + 0.476i)23-s + (0.178 + 0.178i)25-s + (−0.136 + 0.0565i)29-s + 0.611i·31-s + (0.0681 + 0.164i)35-s + (1.57 + 0.651i)37-s + (−0.824 − 0.824i)41-s + (−0.386 − 0.160i)43-s − 1.00i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9583498571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9583498571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.739 + 1.78i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.385 - 0.385i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.36 - 5.70i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (2.30 - 0.956i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + (-1.27 - 3.07i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.28 - 2.28i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.735 - 0.304i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 3.40iT - 31T^{2} \) |
| 37 | \( 1 + (-9.56 - 3.96i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (5.27 + 5.27i)T + 41iT^{2} \) |
| 43 | \( 1 + (2.53 + 1.05i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 + (-7.45 - 3.08i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-6.14 - 2.54i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-2.67 - 6.46i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (10.2 - 4.26i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (6.37 + 6.37i)T + 71iT^{2} \) |
| 73 | \( 1 + (-9.03 + 9.03i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.22T + 79T^{2} \) |
| 83 | \( 1 + (14.8 - 6.14i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.97 - 4.97i)T - 89iT^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−9.901677017760639686234332504770, −9.319223794098781446440932264948, −8.483539827754859508577709314447, −7.48892277180501814413809869759, −6.84511361011379405671265351356, −5.62944011636172327035219551036, −4.87472449605212109521307451054, −4.15924884995910614299443118273, −2.56159310786532401422627820405, −1.64834761477451011530836636229,
0.39305672192717464097494683685, 2.43308530497786493682501975117, 3.03851051420665514429216065759, 4.30167108679816250974097928741, 5.43636782451697961233246905332, 6.25428146599606311324387202661, 6.95417119903729831328691865206, 7.983936096966825899226604276921, 8.676676851256270285986992006694, 9.687650695535099930813489179317