| L(s) = 1 | + (−2.70 − 1.45i)2-s + (−0.133 + 0.127i)3-s + (3.00 + 4.55i)4-s + (4.99 − 5.22i)5-s + (0.546 − 0.150i)6-s + (8.60 − 4.62i)7-s + (−0.400 − 4.45i)8-s + (−0.402 + 8.95i)9-s + (−21.1 + 6.87i)10-s + (1.95 + 14.4i)11-s + (−0.980 − 0.223i)12-s + (0.276 − 0.647i)13-s − 30.0·14-s + 1.33i·15-s + (3.16 − 7.41i)16-s + (10.4 + 11.9i)17-s + ⋯ |
| L(s) = 1 | + (−1.35 − 0.728i)2-s + (−0.0444 + 0.0424i)3-s + (0.751 + 1.13i)4-s + (0.999 − 1.04i)5-s + (0.0911 − 0.0251i)6-s + (1.22 − 0.661i)7-s + (−0.0501 − 0.556i)8-s + (−0.0446 + 0.995i)9-s + (−2.11 + 0.687i)10-s + (0.177 + 1.31i)11-s + (−0.0817 − 0.0186i)12-s + (0.0212 − 0.0498i)13-s − 2.14·14-s + 0.0889i·15-s + (0.198 − 0.463i)16-s + (0.615 + 0.704i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.949548 - 0.483401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949548 - 0.483401i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 + (-101. - 184. i)T \) |
| good | 2 | \( 1 + (2.70 + 1.45i)T + (2.20 + 3.33i)T^{2} \) |
| 3 | \( 1 + (0.133 - 0.127i)T + (0.403 - 8.99i)T^{2} \) |
| 5 | \( 1 + (-4.99 + 5.22i)T + (-1.12 - 24.9i)T^{2} \) |
| 7 | \( 1 + (-8.60 + 4.62i)T + (26.9 - 40.8i)T^{2} \) |
| 11 | \( 1 + (-1.95 - 14.4i)T + (-116. + 32.1i)T^{2} \) |
| 13 | \( 1 + (-0.276 + 0.647i)T + (-116. - 122. i)T^{2} \) |
| 17 | \( 1 + (-10.4 - 11.9i)T + (-38.7 + 286. i)T^{2} \) |
| 19 | \( 1 + (-24.4 - 17.7i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + (17.6 + 24.3i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (1.69 + 0.723i)T + (581. + 607. i)T^{2} \) |
| 31 | \( 1 + (-16.9 - 3.87i)T + (865. + 416. i)T^{2} \) |
| 37 | \( 1 + (-5.30 + 39.1i)T + (-1.31e3 - 364. i)T^{2} \) |
| 41 | \( 1 + (9.35 - 15.6i)T + (-796. - 1.48e3i)T^{2} \) |
| 43 | \( 1 + (-14.1 + 61.8i)T + (-1.66e3 - 802. i)T^{2} \) |
| 47 | \( 1 + (-1.28 - 28.6i)T + (-2.20e3 + 198. i)T^{2} \) |
| 53 | \( 1 + (11.4 - 17.3i)T + (-1.10e3 - 2.58e3i)T^{2} \) |
| 59 | \( 1 + (59.3 + 35.4i)T + (1.64e3 + 3.06e3i)T^{2} \) |
| 61 | \( 1 + (-25.9 + 8.41i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (47.0 - 97.6i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (36.7 + 113. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-18.9 + 23.7i)T + (-1.18e3 - 5.19e3i)T^{2} \) |
| 79 | \( 1 + (71.8 + 6.46i)T + (6.14e3 + 1.11e3i)T^{2} \) |
| 83 | \( 1 + (-35.6 + 109. i)T + (-5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (14.6 - 80.6i)T + (-7.41e3 - 2.78e3i)T^{2} \) |
| 97 | \( 1 + (23.8 - 63.6i)T + (-7.08e3 - 6.19e3i)T^{2} \) |
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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
−11.84482306539070066211556776916, −10.54908030759764680615564000877, −10.15149391459034268675120333774, −9.196599220362013027582814603696, −8.108877119939050162908030737288, −7.58031562968499360377810050658, −5.50682370864675230918194774997, −4.52176913688861500839721514434, −2.00996913575753212211465877381, −1.34211835075676048149077171885,
1.23734861625275467521259759251, 3.04308663835287123167211373239, 5.53118026697560371854872500483, 6.27174208249355933405550659900, 7.32453860871886267687271126406, 8.348537274447769471027581173917, 9.317782934167198148401728744897, 9.927030384253487376911875304376, 11.19877714395607869829068819095, 11.73689272505274783698814418686
