| L(s) = 1 | + (0.743 − 0.669i)2-s + (−0.970 − 0.102i)3-s + (0.104 − 0.994i)4-s + (0.582 − 2.15i)5-s + (−0.789 + 0.573i)6-s + (−2.62 − 0.309i)7-s + (−0.587 − 0.809i)8-s + (−2.00 − 0.425i)9-s + (−1.01 − 1.99i)10-s + (−1.53 + 0.326i)11-s + (−0.202 + 0.954i)12-s + (0.429 − 0.139i)13-s + (−2.15 + 1.52i)14-s + (−0.785 + 2.03i)15-s + (−0.978 − 0.207i)16-s + (2.51 − 5.65i)17-s + ⋯ |
| L(s) = 1 | + (0.525 − 0.473i)2-s + (−0.560 − 0.0589i)3-s + (0.0522 − 0.497i)4-s + (0.260 − 0.965i)5-s + (−0.322 + 0.234i)6-s + (−0.993 − 0.116i)7-s + (−0.207 − 0.286i)8-s + (−0.667 − 0.141i)9-s + (−0.319 − 0.630i)10-s + (−0.462 + 0.0983i)11-s + (−0.0585 + 0.275i)12-s + (0.119 − 0.0386i)13-s + (−0.577 + 0.408i)14-s + (−0.202 + 0.525i)15-s + (−0.244 − 0.0519i)16-s + (0.610 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.255401 - 0.983039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.255401 - 0.983039i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.743 + 0.669i)T \) |
| 5 | \( 1 + (-0.582 + 2.15i)T \) |
| 7 | \( 1 + (2.62 + 0.309i)T \) |
| good | 3 | \( 1 + (0.970 + 0.102i)T + (2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (1.53 - 0.326i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.429 + 0.139i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.51 + 5.65i)T + (-11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.375 - 3.57i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.68 + 1.51i)T + (2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (8.17 + 5.94i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.29 - 2.35i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.23 + 5.79i)T + (-33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.02 - 6.22i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 7.11iT - 43T^{2} \) |
| 47 | \( 1 + (2.00 + 4.51i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-11.5 - 1.21i)T + (51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.41 + 8.23i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-5.67 - 6.29i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (3.65 - 8.22i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-10.5 - 7.67i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.662 + 3.11i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (12.9 - 5.77i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.78 - 5.21i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.72 + 1.91i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.482 + 0.664i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.39040000670502815487095387249, −10.13140913953020905183639593433, −9.542769023892690253451937814501, −8.452617064636329172599409054374, −7.05980804998751412845105266369, −5.78421477423142953719835264425, −5.35369362585445907697521837786, −3.96897221178846647539992883890, −2.62776105055324022652578799297, −0.61366830961884486269484858121,
2.69879367549374592173493583874, 3.66172488893991096019783762423, 5.32994411284377120846652929213, 6.05916702901373148820981416900, 6.78696072655280255147240293332, 7.87860529074268121084667121668, 9.100178206679724579655441284627, 10.24934050824619628605503930022, 10.98309545790491676074114516367, 11.85168249927024153709795976075
