| L(s) = 1 | + (−1.69 − 0.259i)3-s + (−0.724 − 0.859i)5-s + (−2.54 + 3.52i)7-s + (−0.0410 − 0.0128i)9-s + (3.97 − 3.75i)11-s + (0.411 − 3.09i)13-s + (1.00 + 1.64i)15-s + (−3.90 + 5.87i)17-s + (4.29 + 0.993i)19-s + (5.23 − 5.33i)21-s + (1.11 + 5.29i)23-s + (0.633 − 3.68i)25-s + (4.70 + 2.29i)27-s + (0.0318 − 0.150i)29-s + (9.99 − 1.91i)31-s + ⋯ |
| L(s) = 1 | + (−0.981 − 0.149i)3-s + (−0.323 − 0.384i)5-s + (−0.960 + 1.33i)7-s + (−0.0136 − 0.00427i)9-s + (1.19 − 1.13i)11-s + (0.114 − 0.857i)13-s + (0.260 + 0.425i)15-s + (−0.946 + 1.42i)17-s + (0.986 + 0.227i)19-s + (1.14 − 1.16i)21-s + (0.233 + 1.10i)23-s + (0.126 − 0.736i)25-s + (0.904 + 0.441i)27-s + (0.00592 − 0.0280i)29-s + (1.79 − 0.343i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.860086 - 0.00192105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860086 - 0.00192105i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (3.88 - 12.3i)T \) |
| good | 3 | \( 1 + (1.69 + 0.259i)T + (2.86 + 0.894i)T^{2} \) |
| 5 | \( 1 + (0.724 + 0.859i)T + (-0.847 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.54 - 3.52i)T + (-2.21 - 6.64i)T^{2} \) |
| 11 | \( 1 + (-3.97 + 3.75i)T + (0.624 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.411 + 3.09i)T + (-12.5 - 3.40i)T^{2} \) |
| 17 | \( 1 + (3.90 - 5.87i)T + (-6.57 - 15.6i)T^{2} \) |
| 19 | \( 1 + (-4.29 - 0.993i)T + (17.0 + 8.33i)T^{2} \) |
| 23 | \( 1 + (-1.11 - 5.29i)T + (-21.0 + 9.30i)T^{2} \) |
| 29 | \( 1 + (-0.0318 + 0.150i)T + (-26.5 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-9.99 + 1.91i)T + (28.8 - 11.4i)T^{2} \) |
| 37 | \( 1 + (-4.21 + 1.31i)T + (30.4 - 21.0i)T^{2} \) |
| 41 | \( 1 + (-8.59 + 3.41i)T + (29.8 - 28.1i)T^{2} \) |
| 43 | \( 1 + (0.170 + 0.0604i)T + (33.4 + 27.0i)T^{2} \) |
| 47 | \( 1 + (-0.0288 + 0.114i)T + (-41.4 - 22.2i)T^{2} \) |
| 53 | \( 1 + (-0.180 - 0.135i)T + (14.8 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-0.557 - 0.838i)T + (-22.8 + 54.4i)T^{2} \) |
| 61 | \( 1 + (0.548 + 0.702i)T + (-14.8 + 59.1i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.67i)T + (-11.3 - 66.0i)T^{2} \) |
| 71 | \( 1 + (-0.759 - 8.00i)T + (-69.7 + 13.3i)T^{2} \) |
| 73 | \( 1 + (0.447 - 0.798i)T + (-38.0 - 62.2i)T^{2} \) |
| 79 | \( 1 + (-5.29 - 0.200i)T + (78.7 + 5.97i)T^{2} \) |
| 83 | \( 1 + (-2.99 + 0.813i)T + (71.6 - 41.9i)T^{2} \) |
| 89 | \( 1 + (6.10 - 9.99i)T + (-40.5 - 79.2i)T^{2} \) |
| 97 | \( 1 + (-17.0 - 3.26i)T + (90.1 + 35.8i)T^{2} \) |
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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
−10.71552935461115042939579087527, −9.540583849113283407453583368114, −8.803703562209709193999698940884, −8.098003260903129094156891252156, −6.47167371298347054350636620124, −6.06893194607726149798134555038, −5.41164044317854331073755597746, −3.93100451711688649737424477525, −2.84461421189305055348860472901, −0.858815155172977974440562923396,
0.810757873760280102566277435758, 2.89426652202148348740525604449, 4.24721495391365936296981812285, 4.75474312452648907426907265386, 6.46433731231916750524535752014, 6.73110005672115140465202487486, 7.48937139967439088706312367855, 9.126720598288448826260015211149, 9.726461100450548060243034445494, 10.59081259453827155749431702757
