L(s) = 1 | + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.955 − 0.294i)9-s + (−0.0747 − 0.997i)12-s + (0.277 + 1.21i)13-s + (−0.988 − 0.149i)16-s + (−0.222 − 0.385i)19-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.134 − 1.79i)37-s + (0.455 + 1.16i)39-s + (−0.277 + 0.347i)43-s − 0.999·48-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.955 − 0.294i)9-s + (−0.0747 − 0.997i)12-s + (0.277 + 1.21i)13-s + (−0.988 − 0.149i)16-s + (−0.222 − 0.385i)19-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (−0.900 + 1.56i)31-s + (−0.222 − 0.974i)36-s + (−0.134 − 1.79i)37-s + (0.455 + 1.16i)39-s + (−0.277 + 0.347i)43-s − 0.999·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.429998272\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429998272\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 5 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.134 - 1.79i)T + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (1.32 + 1.22i)T + (0.0747 + 0.997i)T^{2} \) |
| 79 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 97 | \( 1 - 0.445T + T^{2} \) |
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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
−9.958423945781953816511317805704, −9.063830254772498262853318415111, −8.731545780305191486167288205783, −7.41658996441832866956035723087, −6.78360082899350540450700135392, −5.86222274899234477911749416302, −4.68021302960382320095835814905, −3.82948566011746862394448093722, −2.42961893399633884457138482599, −1.52269818556770719664296549277,
1.98063021025792868020150709312, 3.14343118943106685225521724697, 3.71505156662742406021439694948, 4.79154628753446788096683832677, 6.06924805532997548745340847968, 7.24875808115609419188296940210, 7.899132972466013267170285492071, 8.402985542660591258671358170841, 9.313576941888436956238121800848, 10.07392279611246063342360701706