| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s + 17-s − 18-s + 19-s − 21-s + 22-s + 23-s + 24-s − 26-s − 27-s + 28-s + 29-s − 31-s − 32-s + 33-s − 34-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 11-s − 12-s + 13-s − 14-s + 16-s + 17-s − 18-s + 19-s − 21-s + 22-s + 23-s + 24-s − 26-s − 27-s + 28-s + 29-s − 31-s − 32-s + 33-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.063947717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063947717\) |
| \(L(1)\) |
\(\approx\) |
\(0.6669537020\) |
| \(L(1)\) |
\(\approx\) |
\(0.6669537020\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 71 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 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 | \( 1 + T \) |
| 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
−24.57840842543663620980322021486, −23.67812502515165574712496745359, −23.14397992154699273667799612333, −21.60176788393715812922021964269, −21.031392318801757958335201795, −20.25382474236957017248693311229, −18.65065304832923941814297723971, −18.425409841967655885895828653683, −17.54029383839553628207736164887, −16.70406385324697961128978464654, −15.87855230517349733187292238665, −15.10986364316211564075629743517, −13.68593903576668600138689538193, −12.37438069382010434258821476008, −11.572541800841104418217641671904, −10.76249464527124061576306865729, −10.15616424425269660173980995458, −8.82267918264513978684145252948, −7.81568392738887713930575900922, −7.03419968801515207695227244875, −5.73759045370357971592497078579, −5.04181212103431970735192176825, −3.30301309682346366238632213660, −1.67611455226919366070105133274, −0.78148224364950926570356848966,
0.78148224364950926570356848966, 1.67611455226919366070105133274, 3.30301309682346366238632213660, 5.04181212103431970735192176825, 5.73759045370357971592497078579, 7.03419968801515207695227244875, 7.81568392738887713930575900922, 8.82267918264513978684145252948, 10.15616424425269660173980995458, 10.76249464527124061576306865729, 11.572541800841104418217641671904, 12.37438069382010434258821476008, 13.68593903576668600138689538193, 15.10986364316211564075629743517, 15.87855230517349733187292238665, 16.70406385324697961128978464654, 17.54029383839553628207736164887, 18.425409841967655885895828653683, 18.65065304832923941814297723971, 20.25382474236957017248693311229, 21.031392318801757958335201795, 21.60176788393715812922021964269, 23.14397992154699273667799612333, 23.67812502515165574712496745359, 24.57840842543663620980322021486
