| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.877755738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877755738\) |
| \(L(1)\) |
\(\approx\) |
\(1.132835953\) |
| \(L(1)\) |
\(\approx\) |
\(1.132835953\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2557 | \( 1 \) |
| 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
−19.60437210626979167236588796413, −18.54716156094938397138697745289, −18.305873042911115682225930163344, −17.423388266273574797236773776988, −16.43699328305703638196163030835, −15.92210679399691556722463571312, −15.03527097267797298705529008403, −14.816628629130935308775976952553, −13.83893542021966127287902194743, −12.94561438906060313901040869454, −11.88104135462107296121845605354, −11.3913486467663232224304858676, −10.80778911212794565111609973780, −9.77893501578325179201024352551, −8.90851137411109668547551005098, −8.51836586460964053230176545967, −7.97630798987089795996131412672, −7.03916783777912482868418635510, −6.69150343040606620737936126529, −5.20944913637347047846780760666, −4.172113710842152218700826244197, −3.51000540579626862300503171434, −2.65102069296346843896328768532, −1.54404532891242607329457580871, −1.016557011659931342400813744530,
1.016557011659931342400813744530, 1.54404532891242607329457580871, 2.65102069296346843896328768532, 3.51000540579626862300503171434, 4.172113710842152218700826244197, 5.20944913637347047846780760666, 6.69150343040606620737936126529, 7.03916783777912482868418635510, 7.97630798987089795996131412672, 8.51836586460964053230176545967, 8.90851137411109668547551005098, 9.77893501578325179201024352551, 10.80778911212794565111609973780, 11.3913486467663232224304858676, 11.88104135462107296121845605354, 12.94561438906060313901040869454, 13.83893542021966127287902194743, 14.816628629130935308775976952553, 15.03527097267797298705529008403, 15.92210679399691556722463571312, 16.43699328305703638196163030835, 17.423388266273574797236773776988, 18.305873042911115682225930163344, 18.54716156094938397138697745289, 19.60437210626979167236588796413
