L(s) = 1 | + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (0.792 − 2.09i)5-s + 2.44i·6-s + (−1.5 + 2.59i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (1.99 + 2.44i)10-s + (2.12 − 3.67i)11-s + (−2.99 − 1.73i)12-s + (2.12 − 1.22i)13-s + (−2.12 − 3.67i)14-s + (−0.621 − 3.82i)15-s + (−2.00 + 3.46i)16-s + 1.41·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)2-s + (0.866 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.354 − 0.935i)5-s + 0.999i·6-s + (−0.566 + 0.981i)7-s + 0.999·8-s + (0.5 − 0.866i)9-s + (0.632 + 0.774i)10-s + (0.639 − 1.10i)11-s + (−0.866 − 0.500i)12-s + (0.588 − 0.339i)13-s + (−0.566 − 0.981i)14-s + (−0.160 − 0.987i)15-s + (−0.500 + 0.866i)16-s + 0.342·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18243 - 0.00792984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18243 - 0.00792984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.792 + 2.09i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.34iT - 19T^{2} \) |
| 23 | \( 1 + (4.5 - 2.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.36 - 3.67i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.44iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 - 2.59i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 2.59i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.66iT - 89T^{2} \) |
| 97 | \( 1 + (-12.7 - 7.34i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.85193508997812214627072202289, −11.99022971525955601601015163872, −10.16385734055434550373558540963, −9.175443296019755223171705814247, −8.630964985319796978631497564217, −7.84004747951292344976554945082, −6.21892929127647967450222932393, −5.67776004011227768174042298481, −3.66977535277130217605698550385, −1.49637292456677036781826674845,
2.14378820246974677509595741166, 3.48054605144381372435472814821, 4.38427260930980732979795789538, 6.81676665586777729000675347994, 7.59856246335112219492338031685, 9.065888561669418752668682400240, 9.757577725312394277365377400484, 10.48284523038252874229390751607, 11.31896373644400899649378065772, 12.76047389329948893842270411304