| L(s) = 1 | + (−0.123 + 0.255i)2-s + (1.44 + 0.957i)3-s + (1.19 + 1.50i)4-s + (1.22 + 0.378i)5-s + (−0.422 + 0.250i)6-s + (−2.64 − 0.0636i)7-s + (−1.08 + 0.247i)8-s + (1.16 + 2.76i)9-s + (−0.247 + 0.266i)10-s + (−3.82 + 0.286i)11-s + (0.289 + 3.31i)12-s + (4.22 − 0.316i)13-s + (0.341 − 0.667i)14-s + (1.40 + 1.71i)15-s + (−0.784 + 3.43i)16-s + (0.900 + 2.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.0869 + 0.180i)2-s + (0.833 + 0.553i)3-s + (0.598 + 0.750i)4-s + (0.548 + 0.169i)5-s + (−0.172 + 0.102i)6-s + (−0.999 − 0.0240i)7-s + (−0.383 + 0.0874i)8-s + (0.388 + 0.921i)9-s + (−0.0782 + 0.0843i)10-s + (−1.15 + 0.0864i)11-s + (0.0835 + 0.956i)12-s + (1.17 − 0.0878i)13-s + (0.0913 − 0.178i)14-s + (0.363 + 0.444i)15-s + (−0.196 + 0.858i)16-s + (0.218 + 0.556i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33786 + 1.35922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33786 + 1.35922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.44 - 0.957i)T \) |
| 7 | \( 1 + (2.64 + 0.0636i)T \) |
| good | 2 | \( 1 + (0.123 - 0.255i)T + (-1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (-1.22 - 0.378i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (3.82 - 0.286i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (-4.22 + 0.316i)T + (12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-0.900 - 2.29i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.20 + 1.27i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.21 + 8.06i)T + (-21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-5.30 + 2.08i)T + (21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + 0.428iT - 31T^{2} \) |
| 37 | \( 1 + (-8.80 - 1.32i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-0.135 + 0.126i)T + (3.06 - 40.8i)T^{2} \) |
| 43 | \( 1 + (-5.21 - 4.83i)T + (3.21 + 42.8i)T^{2} \) |
| 47 | \( 1 + (10.4 + 5.05i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-0.241 - 1.60i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-0.772 + 3.38i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-3.12 - 2.48i)T + (13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 + (-3.06 + 2.44i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (-7.31 - 0.548i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 - 7.35T + 79T^{2} \) |
| 83 | \( 1 + (-0.00990 + 0.132i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (12.0 + 8.19i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (13.9 + 8.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.07852120386877687555505143905, −10.32633907877873837062330573267, −9.608204544366392891242679866277, −8.426187127874302329327475687188, −7.973486019104206414857766344730, −6.69558417854602460522862913164, −5.90033582087144467018605020169, −4.27331665232249669267675146880, −3.12356521571211388088108266655, −2.42298957887605702662000301649,
1.22772615877637065030277673744, 2.55364060238828750683505358664, 3.48289295804219429445000894843, 5.49119420914765228007318865418, 6.17946616906655599739827333725, 7.17908641740473629262459915002, 8.125029178365845815798147711857, 9.493715996271348720363153963828, 9.657317104743648595824311884701, 10.78814164606688869822959174507
