L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s − 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s − 22-s − 23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.963929616\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.963929616\) |
\(L(1)\) |
\(\approx\) |
\(2.517852251\) |
\(L(1)\) |
\(\approx\) |
\(2.517852251\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1033 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 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
−21.45808382779393398213227846294, −20.67457407969200477224594835670, −20.33174312133962866920159905894, −19.43244490890856913209032534802, −18.77154690479651963394004286002, −17.76396403603661915339303261028, −16.40246563763102931932098166676, −15.85755531953847012337881879946, −15.00352710771884096250618920707, −14.54419794513183726709919201147, −13.87428688274220257027142141237, −12.868772019265497311226088206841, −12.167368479151846916339151943681, −11.45407698832281336874744781153, −10.487172839864718332618844564739, −9.61033125669792573566200941329, −8.11895886409747040562712658966, −7.728279079810200086134797658569, −7.28552194238342835712097160231, −5.71768142066804416362431698172, −4.77062758128312056759948333089, −4.19469010108334357398078023047, −3.13297778280283103907163159543, −2.496190219884075234644007637956, −1.32333741042229183244633326411,
1.32333741042229183244633326411, 2.496190219884075234644007637956, 3.13297778280283103907163159543, 4.19469010108334357398078023047, 4.77062758128312056759948333089, 5.71768142066804416362431698172, 7.28552194238342835712097160231, 7.728279079810200086134797658569, 8.11895886409747040562712658966, 9.61033125669792573566200941329, 10.487172839864718332618844564739, 11.45407698832281336874744781153, 12.167368479151846916339151943681, 12.868772019265497311226088206841, 13.87428688274220257027142141237, 14.54419794513183726709919201147, 15.00352710771884096250618920707, 15.85755531953847012337881879946, 16.40246563763102931932098166676, 17.76396403603661915339303261028, 18.77154690479651963394004286002, 19.43244490890856913209032534802, 20.33174312133962866920159905894, 20.67457407969200477224594835670, 21.45808382779393398213227846294