L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 13-s + 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s − 40-s − 41-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 11-s + 13-s + 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s − 26-s − 28-s + 29-s + 31-s − 32-s + 34-s − 35-s + 37-s − 38-s − 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8943492599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8943492599\) |
\(L(1)\) |
\(\approx\) |
\(0.7803375671\) |
\(L(1)\) |
\(\approx\) |
\(0.7803375671\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 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
−25.00838232172142336370200351237, −23.83507987693949191626892460496, −22.79861990417864498033240088315, −21.698953186549048195902362376465, −20.901454979325997176034772560121, −20.14476442633885400379801642691, −19.11887961327711160020380270558, −18.236987357801661269835314354654, −17.748099764059375885436938951013, −16.586104507051073190632546478469, −15.95459493960004047161312700510, −15.09903729084220708575940202249, −13.52287995634325006554401921380, −13.09021323039172953264574302077, −11.72319468806766554493591274178, −10.61268941531571974408527304379, −9.984619317369109937372823227589, −9.112943285857401662805664573546, −8.25980328153644861598775199251, −6.874006161944720562074328233938, −6.27916480359486153705408477897, −5.176820857680449693761216473450, −3.218468865743940311565412297516, −2.41456870050531142193684063387, −0.99979569085501172249760102723,
0.99979569085501172249760102723, 2.41456870050531142193684063387, 3.218468865743940311565412297516, 5.176820857680449693761216473450, 6.27916480359486153705408477897, 6.874006161944720562074328233938, 8.25980328153644861598775199251, 9.112943285857401662805664573546, 9.984619317369109937372823227589, 10.61268941531571974408527304379, 11.72319468806766554493591274178, 13.09021323039172953264574302077, 13.52287995634325006554401921380, 15.09903729084220708575940202249, 15.95459493960004047161312700510, 16.586104507051073190632546478469, 17.748099764059375885436938951013, 18.236987357801661269835314354654, 19.11887961327711160020380270558, 20.14476442633885400379801642691, 20.901454979325997176034772560121, 21.698953186549048195902362376465, 22.79861990417864498033240088315, 23.83507987693949191626892460496, 25.00838232172142336370200351237