| L(s) = 1 | + (0.578 + 1.29i)2-s + (0.987 + 0.156i)3-s + (−1.33 + 1.49i)4-s + (0.979 + 2.01i)5-s + (0.369 + 1.36i)6-s + (0.0808 − 0.0808i)7-s + (−2.69 − 0.853i)8-s + (0.951 + 0.309i)9-s + (−2.02 + 2.42i)10-s + (0.521 − 0.169i)11-s + (−1.54 + 1.26i)12-s + (−0.379 − 0.743i)13-s + (0.151 + 0.0575i)14-s + (0.652 + 2.13i)15-s + (−0.459 − 3.97i)16-s + (0.514 + 3.24i)17-s + ⋯ |
| L(s) = 1 | + (0.409 + 0.912i)2-s + (0.570 + 0.0903i)3-s + (−0.665 + 0.746i)4-s + (0.437 + 0.898i)5-s + (0.150 + 0.557i)6-s + (0.0305 − 0.0305i)7-s + (−0.953 − 0.301i)8-s + (0.317 + 0.103i)9-s + (−0.641 + 0.767i)10-s + (0.157 − 0.0510i)11-s + (−0.446 + 0.365i)12-s + (−0.105 − 0.206i)13-s + (0.0403 + 0.0153i)14-s + (0.168 + 0.552i)15-s + (−0.114 − 0.993i)16-s + (0.124 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01344 + 1.52667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01344 + 1.52667i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.578 - 1.29i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (-0.979 - 2.01i)T \) |
| good | 7 | \( 1 + (-0.0808 + 0.0808i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.521 + 0.169i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.379 + 0.743i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.514 - 3.24i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (2.87 + 2.08i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.38 + 6.63i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-4.57 - 6.29i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 1.90i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.401 + 0.204i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.29 + 3.97i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (2.31 + 2.31i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.528 + 3.33i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-2.22 + 14.0i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.80 + 5.54i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.07 + 6.39i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (12.2 - 1.93i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-9.28 - 12.7i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.31 - 0.670i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-4.21 + 3.06i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.973 - 6.14i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-9.70 + 3.15i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.70 + 0.270i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.40018479590428966961463690718, −10.96092532545636556306975544239, −10.07647359950295551373619868147, −8.940263016519873024182599962706, −8.171488268613812232340867390837, −6.97706902252942726103386548033, −6.37629616500033869707545086737, −5.04964293996731332975479074051, −3.77077190691316390638955682798, −2.60906853804804925726700433139,
1.34799012752577102394020127802, 2.70738493797472134879516342665, 4.13041916539875376124362702129, 5.09223620167832395517463046892, 6.25199510188993458349648647105, 7.85979479187982414008203923735, 8.973937222762602375698309631162, 9.522565626147346209134658012514, 10.44850941407905164975659866729, 11.73488549456362183102764215392
