L(s) = 1 | + (−0.623 + 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (1.62 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 + 0.433i)8-s + (1.40 + 1.75i)9-s + (0.900 − 0.433i)10-s + (−0.400 + 1.75i)12-s + (−0.900 − 0.433i)14-s + (1.12 + 1.40i)15-s + (−0.900 + 0.433i)16-s − 2.24·18-s + (−0.222 + 0.974i)20-s + (0.400 − 1.75i)21-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)2-s + (−1.62 − 0.781i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (1.62 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 + 0.433i)8-s + (1.40 + 1.75i)9-s + (0.900 − 0.433i)10-s + (−0.400 + 1.75i)12-s + (−0.900 − 0.433i)14-s + (1.12 + 1.40i)15-s + (−0.900 + 0.433i)16-s − 2.24·18-s + (−0.222 + 0.974i)20-s + (0.400 − 1.75i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2808149896\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2808149896\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (-0.222 - 0.974i)T \) |
good | 3 | \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \) |
| 53 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 61 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 + 2T + T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.277 - 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.24063302857148971333138762387, −8.934767566004377191802902626094, −8.268402824316746930758720506158, −7.44191489209069009029912107714, −6.66424095516793258767757267778, −5.89869047706462077277749229757, −5.16251400027676591725366808021, −4.41152545774057166764440618238, −1.99739946321435648735086389024, −0.50068067144684696359766649551,
1.12227874522549266783737319679, 3.32776932722988761998532540699, 4.12075175324556659238615922478, 4.76895444294960565116112356973, 6.05615879241325725409381191693, 7.20004402688021694384221563091, 7.63952502730630920954170155093, 8.969351617873892448132998839090, 9.980612453167528991655488616843, 10.43919929179297851213173250094