| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 743 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 743 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.250353005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.250353005\) |
| \(L(1)\) |
\(\approx\) |
\(2.420330984\) |
| \(L(1)\) |
\(\approx\) |
\(2.420330984\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 743 | \( 1 \) |
| 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
−22.4432960710028334452536690657, −21.60073208657907313489065947153, −20.48752543624963789830241649675, −19.782465569667902162465758428067, −19.55005307363749087659556661320, −18.69352058859078764260103361907, −17.0622357954359691247420449329, −16.22100216339202438298234290371, −15.45888854169028776589502991052, −14.941416023493342273437436577377, −14.10216962738237918001626657081, −13.24016297673806730719922558942, −12.505746504101740077881762630243, −11.801183010858324429410243678196, −10.7951667764463409737197608983, −9.60488984726875949128948604603, −8.897931984462569114205606597958, −7.46070028121591421172084172400, −7.20602870531219131028789603216, −6.14237788756544997292809711763, −4.645862274448745496332587814491, −4.03500763361550251365412537101, −3.14929293978140764594618850312, −2.48110565602224358766064286985, −0.95076462210110645699333903891,
0.95076462210110645699333903891, 2.48110565602224358766064286985, 3.14929293978140764594618850312, 4.03500763361550251365412537101, 4.645862274448745496332587814491, 6.14237788756544997292809711763, 7.20602870531219131028789603216, 7.46070028121591421172084172400, 8.897931984462569114205606597958, 9.60488984726875949128948604603, 10.7951667764463409737197608983, 11.801183010858324429410243678196, 12.505746504101740077881762630243, 13.24016297673806730719922558942, 14.10216962738237918001626657081, 14.941416023493342273437436577377, 15.45888854169028776589502991052, 16.22100216339202438298234290371, 17.0622357954359691247420449329, 18.69352058859078764260103361907, 19.55005307363749087659556661320, 19.782465569667902162465758428067, 20.48752543624963789830241649675, 21.60073208657907313489065947153, 22.4432960710028334452536690657
