| L(s) = 1 | + (−0.510 + 0.294i)2-s + (−1.38 + 2.66i)3-s + (−1.82 + 3.16i)4-s + (4.01 + 2.31i)5-s + (−0.0765 − 1.76i)6-s + (2.98 + 5.16i)7-s − 4.51i·8-s + (−5.15 − 7.37i)9-s − 2.73·10-s + (−5.45 + 3.15i)11-s + (−5.88 − 9.24i)12-s + (−0.650 + 1.12i)13-s + (−3.04 − 1.75i)14-s + (−11.7 + 7.46i)15-s + (−5.97 − 10.3i)16-s + 4.12i·17-s + ⋯ |
| L(s) = 1 | + (−0.255 + 0.147i)2-s + (−0.462 + 0.886i)3-s + (−0.456 + 0.790i)4-s + (0.802 + 0.463i)5-s + (−0.0127 − 0.294i)6-s + (0.425 + 0.737i)7-s − 0.564i·8-s + (−0.572 − 0.819i)9-s − 0.273·10-s + (−0.496 + 0.286i)11-s + (−0.490 − 0.770i)12-s + (−0.0500 + 0.0866i)13-s + (−0.217 − 0.125i)14-s + (−0.781 + 0.497i)15-s + (−0.373 − 0.646i)16-s + 0.242i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.119121 + 0.902182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119121 + 0.902182i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.38 - 2.66i)T \) |
| 17 | \( 1 - 4.12iT \) |
| good | 2 | \( 1 + (0.510 - 0.294i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + (-4.01 - 2.31i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-2.98 - 5.16i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (5.45 - 3.15i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.650 - 1.12i)T + (-84.5 - 146. i)T^{2} \) |
| 19 | \( 1 + 25.8T + 361T^{2} \) |
| 23 | \( 1 + (-24.7 - 14.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-21.3 + 12.3i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (7.27 - 12.6i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 36.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-42.5 - 24.5i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-19.2 - 33.3i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (13.8 - 7.98i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 23.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-7.88 - 4.55i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.8 - 58.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.7 + 30.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 120. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-67.7 - 117. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-51.8 + 29.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 71.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (77.2 + 133. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.08585803062640169626496884495, −12.18174344621761578202341875180, −11.04631489873259832546391147103, −10.07526003783509229092931169136, −9.150384258339305534084068602158, −8.277926362154737923056296551072, −6.71153123975541271558981321672, −5.48682117960995915859923746732, −4.32373235896141920244835764218, −2.71773170227929291937056496818,
0.67707741634051250928819369732, 2.03178704243290884822064016597, 4.76881375173715851994675769536, 5.65655064854318940417544600757, 6.80294737994824564710002729918, 8.192998349727191133991941227227, 9.155816782293191898843681242523, 10.55532934598926797366022812461, 10.91876925208366732698011650904, 12.47247678970647031975319052340
