L(s) = 1 | + (−0.173 + 0.300i)2-s + (1.11 − 1.32i)3-s + (0.939 + 1.62i)4-s + (−0.326 − 0.565i)5-s + (0.205 + 0.565i)6-s + (−0.266 + 0.460i)7-s − 1.34·8-s + (−0.520 − 2.95i)9-s + 0.226·10-s + (0.5 − 0.866i)11-s + (3.20 + 0.565i)12-s + (1.15 + 1.99i)13-s + (−0.0923 − 0.160i)14-s + (−1.11 − 0.196i)15-s + (−1.64 + 2.84i)16-s − 4.41·17-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.212i)2-s + (0.642 − 0.766i)3-s + (0.469 + 0.813i)4-s + (−0.145 − 0.252i)5-s + (0.0839 + 0.230i)6-s + (−0.100 + 0.174i)7-s − 0.476·8-s + (−0.173 − 0.984i)9-s + 0.0716·10-s + (0.150 − 0.261i)11-s + (0.925 + 0.163i)12-s + (0.319 + 0.553i)13-s + (−0.0246 − 0.0427i)14-s + (−0.287 − 0.0506i)15-s + (−0.411 + 0.712i)16-s − 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15669\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.11 + 1.32i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.173 - 0.300i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.326 + 0.565i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.266 - 0.460i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-1.15 - 1.99i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 + (0.705 + 1.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.17 - 5.50i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.08 - 1.87i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 + (-4.65 - 8.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.14 + 10.6i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.93 + 3.34i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + (-1.61 - 2.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.26 + 7.38i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.48 + 4.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.98T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + (-2.43 + 4.22i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.29 + 3.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + (6.32 - 10.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.77745100959648994204920313693, −12.77877194568126969366914715362, −12.09049015043492013796127876828, −10.93055533624928347816825680049, −8.936597985983079977988058427238, −8.498515348238877535176715600957, −7.16576415121361910830222974071, −6.32538323149485518983050714305, −3.95841629004373223610675564355, −2.37101318364925647059534818824,
2.42509581480708884265128443841, 4.13069803706824618099025629107, 5.72053030813267742374670059799, 7.17809891426627090209318555779, 8.676951554736458753380725547181, 9.713412905019593754802042315726, 10.66427037100759327767608180060, 11.33811641512491537451562548090, 13.01804083516216763878517296488, 14.11233409881310068474863605086