L(s) = 1 | + (0.587 + 0.809i)2-s + (0.863 + 0.280i)3-s + (−0.309 + 0.951i)4-s + (−0.156 + 0.987i)5-s + (0.280 + 0.863i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (−0.142 − 0.103i)9-s + (−0.891 + 0.453i)10-s + (−0.533 + 0.734i)12-s + (1.04 − 1.44i)13-s + (−0.809 + 0.587i)14-s + (−0.412 + 0.809i)15-s + (−0.809 − 0.587i)16-s − 0.175i·18-s + (−0.437 − 1.34i)19-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)2-s + (0.863 + 0.280i)3-s + (−0.309 + 0.951i)4-s + (−0.156 + 0.987i)5-s + (0.280 + 0.863i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (−0.142 − 0.103i)9-s + (−0.891 + 0.453i)10-s + (−0.533 + 0.734i)12-s + (1.04 − 1.44i)13-s + (−0.809 + 0.587i)14-s + (−0.412 + 0.809i)15-s + (−0.809 − 0.587i)16-s − 0.175i·18-s + (−0.437 − 1.34i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.743406245\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.743406245\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (0.156 - 0.987i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (-0.863 - 0.280i)T + (0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.437 + 1.34i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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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
−9.764355883517378783119242513692, −8.951190813012240682821858101641, −8.425871026595789518066969352425, −7.65761748104023727048479332378, −6.78019002166400121916917146704, −5.89440184407941572637292602826, −5.27277020796511396516556789809, −3.88062347221887962136328289255, −3.10257271838483219606383702998, −2.63828586661358771614611818293,
1.23065065796331416808026768510, 2.13390926212324365417884007073, 3.51526723981625908947369938262, 4.12361852302141855469205038704, 4.91405983871554474482913259079, 6.09088616887626426564434485704, 6.96714884545689417967051739348, 8.198141441871708149736001086826, 8.689656797300675080027467992540, 9.393359129947616975970819074507