L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 25-s − 27-s + 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s − 57-s + ⋯ |
L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s + 25-s − 27-s + 29-s − 31-s − 33-s − 35-s − 37-s − 39-s + 41-s + 43-s − 45-s − 47-s + 49-s + 51-s − 53-s − 55-s − 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7475038909\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7475038909\) |
\(L(1)\) |
\(\approx\) |
\(0.8071106504\) |
\(L(1)\) |
\(\approx\) |
\(0.8071106504\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 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
−30.56546016175685802216170537662, −29.242823378487636721827334521395, −28.006455842521963452729555056551, −27.50463251533811286246445992468, −26.54607826599937009595547350927, −24.70249611236044990043694110744, −23.99145708958307193176741489049, −22.994177604884986285126178705588, −22.12474197925929969063632827160, −20.84537689176568927077148588205, −19.67494936955169080576522087599, −18.35144023819853553867980194098, −17.53328555413036545501775757087, −16.267668943364381410263844552361, −15.41751803819302293367611100153, −14.03440772808988137627006644797, −12.42016655278868888457002496931, −11.43829100805393573820535556358, −10.880488562437947285332371309965, −8.99041044322388354298083545149, −7.63725952967221761670663025449, −6.426680777461463525192446696654, −4.90215948742937868411286451809, −3.857973350417766331784296841016, −1.28645787889364417182140347191,
1.28645787889364417182140347191, 3.857973350417766331784296841016, 4.90215948742937868411286451809, 6.426680777461463525192446696654, 7.63725952967221761670663025449, 8.99041044322388354298083545149, 10.880488562437947285332371309965, 11.43829100805393573820535556358, 12.42016655278868888457002496931, 14.03440772808988137627006644797, 15.41751803819302293367611100153, 16.267668943364381410263844552361, 17.53328555413036545501775757087, 18.35144023819853553867980194098, 19.67494936955169080576522087599, 20.84537689176568927077148588205, 22.12474197925929969063632827160, 22.994177604884986285126178705588, 23.99145708958307193176741489049, 24.70249611236044990043694110744, 26.54607826599937009595547350927, 27.50463251533811286246445992468, 28.006455842521963452729555056551, 29.242823378487636721827334521395, 30.56546016175685802216170537662