L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s − 34-s + 35-s − 37-s + 38-s − 40-s + 41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 13-s − 14-s + 16-s − 17-s + 19-s − 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s − 34-s + 35-s − 37-s + 38-s − 40-s + 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 993 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 993 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.298453969\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.298453969\) |
\(L(1)\) |
\(\approx\) |
\(1.632588703\) |
\(L(1)\) |
\(\approx\) |
\(1.632588703\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 331 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.09651579295260303834590482857, −20.90168494100649006081262885181, −20.02210896798791955707329603997, −19.469209799631251720286879850672, −19.0917647643996935890222498988, −17.50337606091079620158143381618, −16.74750698544058910323988934366, −15.79934091951732684777436078920, −15.51654228536672549058047029108, −14.49117186685315935726979933659, −13.80996266731870376913348950022, −12.76498869504109557844878763959, −12.19716238760852117781910602010, −11.55936372597722834674274954711, −10.66758249458278981326066102843, −9.627225539182491676221184891, −8.670604256791966654777961800981, −7.349256707525282760102391685687, −6.95129364151737509757438221350, −6.063617069270580501579421734781, −4.840093688952716986516684491795, −4.1910539725255083341821620432, −3.2505479155369903549884766080, −2.574234185465105909101803122360, −0.95905579686796031917785327268,
0.95905579686796031917785327268, 2.574234185465105909101803122360, 3.2505479155369903549884766080, 4.1910539725255083341821620432, 4.840093688952716986516684491795, 6.063617069270580501579421734781, 6.95129364151737509757438221350, 7.349256707525282760102391685687, 8.670604256791966654777961800981, 9.627225539182491676221184891, 10.66758249458278981326066102843, 11.55936372597722834674274954711, 12.19716238760852117781910602010, 12.76498869504109557844878763959, 13.80996266731870376913348950022, 14.49117186685315935726979933659, 15.51654228536672549058047029108, 15.79934091951732684777436078920, 16.74750698544058910323988934366, 17.50337606091079620158143381618, 19.0917647643996935890222498988, 19.469209799631251720286879850672, 20.02210896798791955707329603997, 20.90168494100649006081262885181, 22.09651579295260303834590482857