| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.025441626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025441626\) |
| \(L(1)\) |
\(\approx\) |
\(1.452955364\) |
| \(L(1)\) |
\(\approx\) |
\(1.452955364\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 953 | \( 1 \) |
| 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
−21.68650191565337641439325114435, −21.23757319362063879717884679101, −20.52713736249540989747405257643, −19.55059040705055241740701078834, −18.50634191399529775939593810204, −17.96335943327357429055989572940, −16.6447428459452116782144695545, −16.24133996389143444115563974421, −15.39603122972147355401886016308, −14.79059457018790118343292861566, −13.749706098835473863821322881864, −12.75244833045400841030001551960, −12.21411433457183145538980396407, −11.313485047017126616918146478189, −10.938377906493783707948778711358, −10.14334617716672544878371537603, −8.21761923384515093683139429008, −7.81429200752830211284985220380, −6.772334642454617139396772431730, −5.833478218296566642294652906149, −5.07782255180366584237655928650, −4.312191079218885398015904239874, −3.57237339675458713124286576058, −2.16488559636396210822834353649, −0.98152620631391755757008726291,
0.98152620631391755757008726291, 2.16488559636396210822834353649, 3.57237339675458713124286576058, 4.312191079218885398015904239874, 5.07782255180366584237655928650, 5.833478218296566642294652906149, 6.772334642454617139396772431730, 7.81429200752830211284985220380, 8.21761923384515093683139429008, 10.14334617716672544878371537603, 10.938377906493783707948778711358, 11.313485047017126616918146478189, 12.21411433457183145538980396407, 12.75244833045400841030001551960, 13.749706098835473863821322881864, 14.79059457018790118343292861566, 15.39603122972147355401886016308, 16.24133996389143444115563974421, 16.6447428459452116782144695545, 17.96335943327357429055989572940, 18.50634191399529775939593810204, 19.55059040705055241740701078834, 20.52713736249540989747405257643, 21.23757319362063879717884679101, 21.68650191565337641439325114435
