| L(s) = 1 | + (−1.18 − 1.26i)3-s − 0.939i·5-s + 2.52i·7-s + (−0.186 + 2.99i)9-s − 4·11-s + 13-s + (−1.18 + 1.11i)15-s + 4.10i·17-s + 3.46i·19-s + (3.18 − 2.99i)21-s − 4.74·23-s + 4.11·25-s + (4.00 − 3.31i)27-s + 6.63i·31-s + (4.74 + 5.04i)33-s + ⋯ |
| L(s) = 1 | + (−0.684 − 0.728i)3-s − 0.420i·5-s + 0.954i·7-s + (−0.0620 + 0.998i)9-s − 1.20·11-s + 0.277·13-s + (−0.306 + 0.287i)15-s + 0.996i·17-s + 0.794i·19-s + (0.695 − 0.653i)21-s − 0.989·23-s + 0.823·25-s + (0.769 − 0.638i)27-s + 1.19i·31-s + (0.825 + 0.878i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.590075 + 0.438399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590075 + 0.438399i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.18 + 1.26i)T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.939iT - 5T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 4.10iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 6.63iT - 31T^{2} \) |
| 37 | \( 1 - 0.372T + 37T^{2} \) |
| 41 | \( 1 - 5.04iT - 41T^{2} \) |
| 43 | \( 1 + 0.644iT - 43T^{2} \) |
| 47 | \( 1 - 6.37T + 47T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 1.58iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 5.04iT - 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−10.73726652625496611192785143907, −10.19717308269657106769147203776, −8.753521467706912240853401485255, −8.210173210913777280728936341932, −7.27374697621758542897044588062, −5.99033208842714428815871631204, −5.61489229280164251144276009042, −4.49596179324235228393752801921, −2.77742923917552808008847286173, −1.54764993868016441930613743481,
0.44279854441306119204232679169, 2.72142193057134018236364805470, 3.92628034132890433189917658334, 4.85199291150316994550886777377, 5.77698602242388482684712922595, 6.87842110523183093411997414291, 7.60039063483753335723925466109, 8.838062083774490814825821015189, 9.893055368518671768389348834245, 10.44673385243211086852850687238
