| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (−55.4 − 18.0i)5-s + (111. + 152. i)7-s + (51.7 + 37.6i)8-s + 233. i·10-s + (−373. + 147. i)11-s + (538. − 175. i)13-s + (444. − 611. i)14-s + (79.1 − 243. i)16-s + (78.2 − 240. i)17-s + (675. − 929. i)19-s + (887. − 288. i)20-s + (1.02e3 + 1.23e3i)22-s − 516. i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.992 − 0.322i)5-s + (0.856 + 1.17i)7-s + (0.286 + 0.207i)8-s + 0.737i·10-s + (−0.930 + 0.366i)11-s + (0.884 − 0.287i)13-s + (0.605 − 0.833i)14-s + (0.0772 − 0.237i)16-s + (0.0656 − 0.202i)17-s + (0.429 − 0.590i)19-s + (0.496 − 0.161i)20-s + (0.449 + 0.545i)22-s − 0.203i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6113628798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6113628798\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (373. - 147. i)T \) |
| good | 5 | \( 1 + (55.4 + 18.0i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-111. - 152. i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-538. + 175. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-78.2 + 240. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-675. + 929. i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + 516. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (2.50e3 - 1.82e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (433. + 1.33e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.26e3 - 916. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.61e4 + 1.17e4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 7.77e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.03e4 + 1.41e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.94e4 + 6.31e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.31e4 + 1.81e4i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (4.58e4 + 1.49e4i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + 1.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.56e4 + 5.09e3i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (4.25e4 + 5.85e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (4.33e4 - 1.40e4i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-663. + 2.04e3i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 3.04e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.16e4 + 3.59e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.36062519351081708367680566405, −10.38632363960365392487659327126, −8.977951589912968705701913301334, −8.345599825892550829548032983604, −7.45654712340810366084874622918, −5.55094793166895297086286968011, −4.61378902820648744086109387239, −3.20487816261374194958200797957, −1.86376645543843518059157981632, −0.22467075688637772793785266186,
1.23426987958861394204148835075, 3.52576047234895536714405939256, 4.49739383942966846931631765886, 5.83962426465477739439886689246, 7.28693779761788562097514846151, 7.77819341581690014314493854311, 8.655467878039887878224627523121, 10.23017178619006690121294099141, 10.93964333315952267130952868768, 11.79028300079289873927750521917
