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 − 31-s + 32-s − 33-s + 34-s + 36-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 − 31-s + 32-s − 33-s + 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.355924047\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.355924047\) |
\(L(1)\) |
\(\approx\) |
\(2.112534460\) |
\(L(1)\) |
\(\approx\) |
\(2.112534460\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 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
−28.54922767829952816180214127615, −27.026617753336564083291914632929, −25.72022741852536521143320958066, −25.545890299908216533016608488483, −24.175596775524331985237975845265, −23.42633713057605812623241288835, −22.154119391961161470348551825539, −21.37623735639373007808585462823, −20.35021560452928102121168383404, −19.53433663317930021328106214456, −18.643242579131048899626403827693, −16.73405798219195611290227662703, −15.79069174082328882204154506014, −14.904730650371502563219859313444, −13.96824785886142504137606092022, −12.90839510789151427999957199948, −12.34802635379878228964141107605, −10.519944312444731201006854735189, −9.66724648490132915197815508131, −8.0002874498927029919943743915, −7.08450102970237100576616316991, −5.720593968185968773279873568899, −4.28610382679618847319483149562, −3.12823758263358221313062665549, −2.208003673429540919232161402067,
2.208003673429540919232161402067, 3.12823758263358221313062665549, 4.28610382679618847319483149562, 5.720593968185968773279873568899, 7.08450102970237100576616316991, 8.0002874498927029919943743915, 9.66724648490132915197815508131, 10.519944312444731201006854735189, 12.34802635379878228964141107605, 12.90839510789151427999957199948, 13.96824785886142504137606092022, 14.904730650371502563219859313444, 15.79069174082328882204154506014, 16.73405798219195611290227662703, 18.643242579131048899626403827693, 19.53433663317930021328106214456, 20.35021560452928102121168383404, 21.37623735639373007808585462823, 22.154119391961161470348551825539, 23.42633713057605812623241288835, 24.175596775524331985237975845265, 25.545890299908216533016608488483, 25.72022741852536521143320958066, 27.026617753336564083291914632929, 28.54922767829952816180214127615