| L(s) = 1 | + (−0.418 + 0.566i)2-s + (1.04 + 0.471i)3-s + (0.440 + 1.43i)4-s + (1.34 + 0.350i)5-s + (−0.702 + 0.392i)6-s + (−0.836 + 0.675i)7-s + (−2.32 − 0.821i)8-s + (−1.11 − 1.27i)9-s + (−0.761 + 0.614i)10-s + (0.574 − 1.87i)11-s + (−0.218 + 1.70i)12-s + (2.08 + 1.99i)13-s + (−0.0321 − 0.756i)14-s + (1.23 + 0.998i)15-s + (−1.05 + 0.711i)16-s + (2.62 + 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.296 + 0.400i)2-s + (0.601 + 0.271i)3-s + (0.220 + 0.719i)4-s + (0.601 + 0.156i)5-s + (−0.286 + 0.160i)6-s + (−0.316 + 0.255i)7-s + (−0.822 − 0.290i)8-s + (−0.372 − 0.423i)9-s + (−0.240 + 0.194i)10-s + (0.173 − 0.565i)11-s + (−0.0630 + 0.492i)12-s + (0.578 + 0.554i)13-s + (−0.00858 − 0.202i)14-s + (0.319 + 0.257i)15-s + (−0.263 + 0.177i)16-s + (0.636 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.987679 + 0.716746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.987679 + 0.716746i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 149 | \( 1 + (2.10 - 12.0i)T \) |
| good | 2 | \( 1 + (0.418 - 0.566i)T + (-0.585 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-1.04 - 0.471i)T + (1.98 + 2.25i)T^{2} \) |
| 5 | \( 1 + (-1.34 - 0.350i)T + (4.36 + 2.43i)T^{2} \) |
| 7 | \( 1 + (0.836 - 0.675i)T + (1.47 - 6.84i)T^{2} \) |
| 11 | \( 1 + (-0.574 + 1.87i)T + (-9.11 - 6.15i)T^{2} \) |
| 13 | \( 1 + (-2.08 - 1.99i)T + (0.551 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-2.62 - 1.18i)T + (11.2 + 12.7i)T^{2} \) |
| 19 | \( 1 + (2.57 + 0.219i)T + (18.7 + 3.21i)T^{2} \) |
| 23 | \( 1 + (-4.52 + 4.33i)T + (0.976 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 1.71i)T + (-13.0 + 25.8i)T^{2} \) |
| 31 | \( 1 + (-1.36 + 1.30i)T + (1.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.169 + 0.554i)T + (-30.6 - 20.7i)T^{2} \) |
| 41 | \( 1 + (-1.15 + 0.783i)T + (15.2 - 38.0i)T^{2} \) |
| 43 | \( 1 + (2.68 + 5.32i)T + (-25.5 + 34.5i)T^{2} \) |
| 47 | \( 1 + (6.70 - 0.571i)T + (46.3 - 7.94i)T^{2} \) |
| 53 | \( 1 + (-5.38 + 10.6i)T + (-31.5 - 42.6i)T^{2} \) |
| 59 | \( 1 + (-1.28 - 0.220i)T + (55.6 + 19.6i)T^{2} \) |
| 61 | \( 1 + (4.95 - 6.70i)T + (-17.8 - 58.3i)T^{2} \) |
| 67 | \( 1 + (0.234 - 5.51i)T + (-66.7 - 5.68i)T^{2} \) |
| 71 | \( 1 + (4.70 + 1.22i)T + (61.9 + 34.6i)T^{2} \) |
| 73 | \( 1 + (-2.52 - 11.7i)T + (-66.5 + 30.0i)T^{2} \) |
| 79 | \( 1 + (2.09 - 9.72i)T + (-71.9 - 32.5i)T^{2} \) |
| 83 | \( 1 + (0.653 + 3.03i)T + (-75.6 + 34.1i)T^{2} \) |
| 89 | \( 1 + (4.14 - 10.3i)T + (-64.2 - 61.5i)T^{2} \) |
| 97 | \( 1 + (2.09 - 4.15i)T + (-57.6 - 77.9i)T^{2} \) |
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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
−13.30919022784568004694553085291, −12.28803208943974534165434633834, −11.25499129941918713705616938145, −9.875097035929031400427955848840, −8.851404955272646950853605237471, −8.338428507823488110795102297539, −6.77248933600041420166704299151, −5.95885426018918092184294689873, −3.83438616029535646769549347353, −2.70342827358803196522448714790,
1.63284978896887074617785445523, 3.06275955667921880614367291669, 5.19601468159606890259414089705, 6.28938033028871872493955252944, 7.67023003976616549217704557546, 8.932446954272312384453120597134, 9.768538215805111958781331247913, 10.65034865961400011411583813195, 11.68248436887693252269819901416, 13.03218166061815613484138309265
