| L(s) = 1 | + (−122. + 212. i)3-s + (−1.69e3 + 977. i)5-s + (−3.57e3 + 5.25e3i)7-s + (−2.03e4 − 3.52e4i)9-s + (4.58e4 + 2.64e4i)11-s − 1.92e5i·13-s − 4.80e5i·15-s + (5.05e5 + 2.91e5i)17-s + (4.29e4 + 7.43e4i)19-s + (−6.78e5 − 1.40e6i)21-s + (7.78e5 − 4.49e5i)23-s + (9.33e5 − 1.61e6i)25-s + 5.17e6·27-s − 1.30e6·29-s + (8.22e5 − 1.42e6i)31-s + ⋯ |
| L(s) = 1 | + (−0.876 + 1.51i)3-s + (−1.21 + 0.699i)5-s + (−0.562 + 0.826i)7-s + (−1.03 − 1.79i)9-s + (0.943 + 0.544i)11-s − 1.87i·13-s − 2.45i·15-s + (1.46 + 0.847i)17-s + (0.0756 + 0.130i)19-s + (−0.761 − 1.57i)21-s + (0.579 − 0.334i)23-s + (0.478 − 0.828i)25-s + 1.87·27-s − 0.341·29-s + (0.159 − 0.276i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8440835216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8440835216\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (3.57e3 - 5.25e3i)T \) |
| good | 3 | \( 1 + (122. - 212. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.69e3 - 977. i)T + (9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-4.58e4 - 2.64e4i)T + (1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.92e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (-5.05e5 - 2.91e5i)T + (5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.29e4 - 7.43e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-7.78e5 + 4.49e5i)T + (9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-8.22e5 + 1.42e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (8.73e6 + 1.51e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.29e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 3.51e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (5.60e5 + 9.71e5i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.21e7 - 5.56e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (2.74e7 - 4.74e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.77e7 - 1.02e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (5.78e5 + 3.33e5i)T + (1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.38e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.11e8 - 6.45e7i)T + (2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-5.11e8 + 2.95e8i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-8.81e8 + 5.08e8i)T + (1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 3.81e8iT - 7.60e17T^{2} \) |
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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
−11.96816616312045566582335661477, −10.79724738585924523821151011568, −10.20639797070536766631737359812, −9.083069521000930726902675738601, −7.69752011898912732630780723861, −6.15988866099433956445346179156, −5.24270656901727897654491457340, −3.78920006678075121518128799531, −3.23243512578173433728791775870, −0.42631761811751829999247004987,
0.71437368915340255537605068396, 1.37294552932750831425922380245, 3.55076322200958834431531965634, 4.88757090985073113949540810745, 6.46522047979651458741006460551, 7.11832892681280838990957958224, 8.053428237934758701098334938470, 9.386750127070191759794512612600, 11.24811663965767303568232490086, 11.78406346192259852215458238201
