L(s) = 1 | + (1.77 + 1.02i)2-s + (0.5 − 0.866i)3-s + (1.09 + 1.90i)4-s + 3.35i·5-s + (1.77 − 1.02i)6-s + (−1.94 + 1.12i)7-s + 0.405i·8-s + (−0.499 − 0.866i)9-s + (−3.43 + 5.95i)10-s + (4.27 + 2.46i)11-s + 2.19·12-s − 4.60·14-s + (2.90 + 1.67i)15-s + (1.78 − 3.08i)16-s + (0.455 + 0.789i)17-s − 2.04i·18-s + ⋯ |
L(s) = 1 | + (1.25 + 0.724i)2-s + (0.288 − 0.499i)3-s + (0.549 + 0.951i)4-s + 1.50i·5-s + (0.724 − 0.418i)6-s + (−0.735 + 0.424i)7-s + 0.143i·8-s + (−0.166 − 0.288i)9-s + (−1.08 + 1.88i)10-s + (1.28 + 0.744i)11-s + 0.634·12-s − 1.23·14-s + (0.750 + 0.433i)15-s + (0.445 − 0.771i)16-s + (0.110 + 0.191i)17-s − 0.482i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.12611 + 1.89441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.12611 + 1.89441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.77 - 1.02i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.35iT - 5T^{2} \) |
| 7 | \( 1 + (1.94 - 1.12i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.27 - 2.46i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.455 - 0.789i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.29 - 1.90i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 1.75i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.96 + 3.41i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 + (-7.62 - 4.40i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.00 + 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.14 + 1.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.80iT - 47T^{2} \) |
| 53 | \( 1 - 0.542T + 53T^{2} \) |
| 59 | \( 1 + (-4.08 + 2.35i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.83 + 3.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 0.760i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.05 + 1.18i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 7.41iT - 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 2.30iT - 83T^{2} \) |
| 89 | \( 1 + (8.71 + 5.02i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.9 - 8.06i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.48539989636278238051897861338, −10.16710118503500451637999826842, −9.434821680954320282657975477573, −7.995681696573254577736126700239, −6.95775983339680254543665070081, −6.50529980744778656586472388369, −5.93710724502780166732271676387, −4.27744407789312378467317151520, −3.43310146411995023878919577364, −2.36186860793356474826122172332,
1.30029249377825289363571086986, 3.06674744868005468948860916270, 3.97122200562950366872881727527, 4.67588579498723985676805113366, 5.60524366518547260374924464826, 6.69315058692235888439814166013, 8.403095324759946318279475142182, 8.962720183803426781810798675238, 9.880900882884335347992523311382, 10.98699776268778507898342544332