| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.644562554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644562554\) |
| \(L(1)\) |
\(\approx\) |
\(1.143357241\) |
| \(L(1)\) |
\(\approx\) |
\(1.143357241\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1021 | \( 1 \) |
| 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
−21.51161716809761720919198554579, −20.6202425788228073802104164418, −19.868403334302282203165580961323, −19.1543280742807912537095131147, −18.82253278056184379049863833501, −17.64023264035819657245327183505, −16.94043206958278969818643899782, −16.37465671464016209447281321881, −15.2524855776021251755325999315, −14.593960043583335125344492668717, −13.8756219578811052814866982543, −12.65183123076654817211043650265, −12.33530159906627589934796425147, −10.75962374952728491851066526151, −10.04706267310392629908690510126, −9.363487902505602244811516270566, −9.024369826361748956733794116472, −7.94127394310442838243160300287, −6.85263497690485537415884709454, −6.530905537598793311850302047641, −5.24066251749701069313870964364, −3.71030434124688667051073756459, −2.84079592868128181937308228942, −2.084613318972894472809836956199, −1.06697467016381389907045820736,
1.06697467016381389907045820736, 2.084613318972894472809836956199, 2.84079592868128181937308228942, 3.71030434124688667051073756459, 5.24066251749701069313870964364, 6.530905537598793311850302047641, 6.85263497690485537415884709454, 7.94127394310442838243160300287, 9.024369826361748956733794116472, 9.363487902505602244811516270566, 10.04706267310392629908690510126, 10.75962374952728491851066526151, 12.33530159906627589934796425147, 12.65183123076654817211043650265, 13.8756219578811052814866982543, 14.593960043583335125344492668717, 15.2524855776021251755325999315, 16.37465671464016209447281321881, 16.94043206958278969818643899782, 17.64023264035819657245327183505, 18.82253278056184379049863833501, 19.1543280742807912537095131147, 19.868403334302282203165580961323, 20.6202425788228073802104164418, 21.51161716809761720919198554579
