| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2747 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2747 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.406682545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406682545\) |
| \(L(1)\) |
\(\approx\) |
\(1.078929451\) |
| \(L(1)\) |
\(\approx\) |
\(1.078929451\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| 67 | \( 1 \) |
| 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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 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
−19.10561082079474335590873130771, −18.44607357222420371962692236023, −17.93327176154986057084226434868, −16.860325834334903237618452970120, −16.38984417385868049322459705119, −15.35192830115340544310404068428, −14.99264813225726100457015571716, −14.541837156954757910253960938462, −13.352380395052519810798790707183, −12.642707304489907245774242357211, −11.56264866818029471112358462982, −11.18379211897296035916362752839, −10.54864598558742216085604929014, −9.35020092903090537859140500220, −8.71739811436874646646587554978, −8.4694856390700401622112112699, −7.59701981212289846873901445259, −7.00524091438758781298978232459, −6.240780529401129877415607086789, −4.80270192550813386464012001138, −3.94793805517783340733214078328, −3.377566665785640857380054360269, −2.205020323915749488193934980453, −1.55825174257851037328410217613, −0.683261664907532098024605668905,
0.683261664907532098024605668905, 1.55825174257851037328410217613, 2.205020323915749488193934980453, 3.377566665785640857380054360269, 3.94793805517783340733214078328, 4.80270192550813386464012001138, 6.240780529401129877415607086789, 7.00524091438758781298978232459, 7.59701981212289846873901445259, 8.4694856390700401622112112699, 8.71739811436874646646587554978, 9.35020092903090537859140500220, 10.54864598558742216085604929014, 11.18379211897296035916362752839, 11.56264866818029471112358462982, 12.642707304489907245774242357211, 13.352380395052519810798790707183, 14.541837156954757910253960938462, 14.99264813225726100457015571716, 15.35192830115340544310404068428, 16.38984417385868049322459705119, 16.860325834334903237618452970120, 17.93327176154986057084226434868, 18.44607357222420371962692236023, 19.10561082079474335590873130771
