| L(s) = 1 | + (−0.0152 − 0.00133i)2-s + (0.259 − 1.71i)3-s + (−1.96 − 0.347i)4-s + (−1.79 − 1.33i)5-s + (−0.00623 + 0.0256i)6-s + (0.211 + 0.148i)7-s + (0.0589 + 0.0158i)8-s + (−2.86 − 0.890i)9-s + (0.0255 + 0.0226i)10-s + (1.49 − 4.11i)11-s + (−1.10 + 3.28i)12-s + (0.187 + 2.14i)13-s + (−0.00301 − 0.00253i)14-s + (−2.75 + 2.72i)15-s + (3.75 + 1.36i)16-s + (1.70 − 0.456i)17-s + ⋯ |
| L(s) = 1 | + (−0.0107 − 0.000940i)2-s + (0.150 − 0.988i)3-s + (−0.984 − 0.173i)4-s + (−0.802 − 0.596i)5-s + (−0.00254 + 0.0104i)6-s + (0.0799 + 0.0559i)7-s + (0.0208 + 0.00558i)8-s + (−0.954 − 0.296i)9-s + (0.00806 + 0.00717i)10-s + (0.451 − 1.23i)11-s + (−0.319 + 0.947i)12-s + (0.0519 + 0.594i)13-s + (−0.000806 − 0.000676i)14-s + (−0.710 + 0.703i)15-s + (0.939 + 0.341i)16-s + (0.412 − 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371320 - 0.655339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371320 - 0.655339i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.259 + 1.71i)T \) |
| 5 | \( 1 + (1.79 + 1.33i)T \) |
| good | 2 | \( 1 + (0.0152 + 0.00133i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.211 - 0.148i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.49 + 4.11i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.187 - 2.14i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 0.456i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.91 + 2.83i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.32 + 4.74i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-3.59 + 3.01i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.912 - 5.17i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.837 + 3.12i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.241 + 0.288i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.84 - 8.25i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (2.29 - 3.28i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (8.15 - 8.15i)T - 53iT^{2} \) |
| 59 | \( 1 + (-10.6 + 3.86i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.10 + 11.9i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.82 + 0.160i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.44 - 2.56i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.71 + 13.8i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.98 - 2.37i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.432 - 4.94i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (1.44 + 2.50i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.27 - 1.99i)T + (62.3 - 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.89178367973766949659184075574, −12.04820006627095439654096273427, −11.16762650582269595256183110577, −9.377552401525861663430216424339, −8.565927089686808243850979758217, −7.80896397987949230623754020656, −6.30019434788964941110022430437, −4.95305466320767449820107927191, −3.46520231616269921086079772933, −0.852956346485735508253086455592,
3.37477533811730618454012329370, 4.26250932406589852861673807661, 5.48405703188872945872235090354, 7.44926326862600128005234869369, 8.341236283310622206721015403147, 9.663040308007973263950457817900, 10.17970263272732114073469804803, 11.55909083104328367474968874462, 12.41974402751411048526464810783, 13.86829416956763975275027843831
