L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 5·13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s − 21-s − 5·22-s + 5·23-s + 24-s + 25-s + 5·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.218·21-s − 1.06·22-s + 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9805659510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9805659510\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.26017388761884, −13.85609172806573, −12.97380379121628, −12.29587236047089, −12.14146321668103, −11.70621344637316, −11.10078953407188, −10.72747765535470, −10.09982300971327, −9.552769225093428, −9.139666034259168, −8.621429104358733, −7.962614417192901, −7.414192488604566, −6.889545841851994, −6.658930785385808, −5.803851171060518, −5.243943868167722, −4.560672024724569, −4.196770226467693, −3.225325121778831, −2.782797770751831, −1.685894658667565, −1.300163673929317, −0.4120364724154024,
0.4120364724154024, 1.300163673929317, 1.685894658667565, 2.782797770751831, 3.225325121778831, 4.196770226467693, 4.560672024724569, 5.243943868167722, 5.803851171060518, 6.658930785385808, 6.889545841851994, 7.414192488604566, 7.962614417192901, 8.621429104358733, 9.139666034259168, 9.552769225093428, 10.09982300971327, 10.72747765535470, 11.10078953407188, 11.70621344637316, 12.14146321668103, 12.29587236047089, 12.97380379121628, 13.85609172806573, 14.26017388761884