L(s) = 1 | + (−1.21 + 2.10i)2-s + (−0.979 − 1.69i)3-s + (−1.96 − 3.40i)4-s + (−2.07 + 3.59i)5-s + 4.76·6-s + (−2.28 − 1.33i)7-s + 4.69·8-s + (−0.416 + 0.722i)9-s + (−5.05 − 8.74i)10-s + (−0.336 − 0.582i)11-s + (−3.84 + 6.66i)12-s + (5.59 − 3.19i)14-s + 8.12·15-s + (−1.78 + 3.09i)16-s + (0.0504 + 0.0874i)17-s + (−1.01 − 1.75i)18-s + ⋯ |
L(s) = 1 | + (−0.860 + 1.49i)2-s + (−0.565 − 0.979i)3-s + (−0.981 − 1.70i)4-s + (−0.927 + 1.60i)5-s + 1.94·6-s + (−0.863 − 0.503i)7-s + 1.65·8-s + (−0.138 + 0.240i)9-s + (−1.59 − 2.76i)10-s + (−0.101 − 0.175i)11-s + (−1.11 + 1.92i)12-s + (1.49 − 0.854i)14-s + 2.09·15-s + (−0.446 + 0.773i)16-s + (0.0122 + 0.0211i)17-s + (−0.239 − 0.414i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3254205830\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3254205830\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.28 + 1.33i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.979 + 1.69i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.07 - 3.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.336 + 0.582i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.0504 - 0.0874i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.72 + 4.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.231 - 0.400i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.30T + 29T^{2} \) |
| 31 | \( 1 + (3.61 + 6.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 - 4.83T + 43T^{2} \) |
| 47 | \( 1 + (-1.41 + 2.45i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.17 - 12.4i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.208 + 0.361i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.768 - 1.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.94 - 5.09i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + (-3.19 - 5.54i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.143 - 0.249i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.43T + 83T^{2} \) |
| 89 | \( 1 + (5.02 - 8.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−9.945740712822727716619395967885, −9.049487004953381797383955240884, −7.85690109395364467626688540650, −7.37458004524854288780879776769, −6.87296608872181616863868591112, −6.41506082054531249864660175794, −5.60041971830756538428180830350, −4.01130311955753384911896430123, −2.82872917393531415548410717941, −0.68724285241478148535467494262,
0.35441075093533553163489643274, 1.73850407812394181316531483299, 3.37899039110363918918105730485, 3.95678812329928560251728061076, 4.90832933421610327348190800879, 5.68462019833700273626834317349, 7.44733967842482608425294859008, 8.397529897097840675756398549064, 8.955194800374602024672327292357, 9.639335777694291892679940901692