| L(s) = 1 | + (−1.5 − 0.866i)3-s + (−0.452 + 0.261i)5-s + (6.50 + 2.57i)7-s + (1.5 + 2.59i)9-s + (1.56 − 2.70i)11-s − 0.897i·13-s + 0.905·15-s + (−4.81 − 2.77i)17-s + (4.00 − 2.31i)19-s + (−7.52 − 9.50i)21-s + (8.67 + 15.0i)23-s + (−12.3 + 21.4i)25-s − 5.19i·27-s − 19.3·29-s + (42.7 + 24.6i)31-s + ⋯ |
| L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.0905 + 0.0523i)5-s + (0.929 + 0.368i)7-s + (0.166 + 0.288i)9-s + (0.142 − 0.246i)11-s − 0.0690i·13-s + 0.0603·15-s + (−0.282 − 0.163i)17-s + (0.210 − 0.121i)19-s + (−0.358 − 0.452i)21-s + (0.377 + 0.653i)23-s + (−0.494 + 0.856i)25-s − 0.192i·27-s − 0.668·29-s + (1.37 + 0.796i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.645392419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645392419\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (-6.50 - 2.57i)T \) |
| good | 5 | \( 1 + (0.452 - 0.261i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 2.70i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 0.897iT - 169T^{2} \) |
| 17 | \( 1 + (4.81 + 2.77i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4.00 + 2.31i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.67 - 15.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 19.3T + 841T^{2} \) |
| 31 | \( 1 + (-42.7 - 24.6i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (4.76 + 8.25i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 2.51iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 54.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-40.4 + 23.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-14.9 + 25.8i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-65.7 - 37.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-53.7 + 31.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (11.5 - 19.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 53.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-80.5 - 46.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (4.49 + 7.78i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 9.29iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-23.8 + 13.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 117. iT - 9.40e3T^{2} \) |
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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
−10.53811643193373238982802924384, −9.429305803262671218172826822011, −8.556110338384303459884340196268, −7.66317589409268705713561447014, −6.86322482913335890887839952375, −5.69199702016053658534647130999, −5.05599571773735792806725289192, −3.84391694958869381950588606197, −2.35981585884305605985638213438, −1.06198980512104460659716111273,
0.818438142706325462679552611531, 2.31444780857791526685578279235, 3.99699313024408598836877436584, 4.63975930409230720467376066907, 5.69024954575613026659112673797, 6.68060221855019445152032660531, 7.67776141851407598672712255002, 8.474929827622751197619230054472, 9.518970611683322804751474514003, 10.38274614178158736866581131707
