L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.5 − 0.866i)7-s + 0.999·8-s − 6·11-s + (−2.5 − 4.33i)13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (3 + 5.19i)22-s − 6·23-s − 5·25-s + (−2.5 + 4.33i)26-s + (−0.499 + 2.59i)28-s + (3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.944 − 0.327i)7-s + 0.353·8-s − 1.80·11-s + (−0.693 − 1.20i)13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (0.639 + 1.10i)22-s − 1.25·23-s − 25-s + (−0.490 + 0.849i)26-s + (−0.0944 + 0.490i)28-s + (0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6533077557\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6533077557\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.800126945367223941535730406455, −8.302615108749417880550665931833, −8.008131415707988281563912985988, −7.40282711257185787720197550572, −5.76931195722910655549769074273, −5.12998615067236608923242002615, −4.07328144093397480858179774440, −2.87934063903172108645609724631, −1.93404512240286852963362138812, −0.30682151304072432538144532853,
1.68819024478826595079496178551, 2.84055454104471899862961679408, 4.48867585967093749663461055931, 5.16058731088957680612920971133, 5.88254054680190165166433383901, 7.13317487854057583238249128023, 7.83662499241586706486499338570, 8.242833893349697339424528780741, 9.460546799018121678264758689516, 9.958551526223595855206332710371