| L(s) = 1 | − 3.42·5-s + 3.44·7-s + 3.41·11-s − 2.00·13-s + 3.07·17-s − 0.565·19-s − 7.40·23-s + 6.76·25-s − 9.73·29-s − 2.62·31-s − 11.8·35-s − 0.527·37-s − 3.19·41-s + 7.98·43-s + 6.96·47-s + 4.84·49-s − 2.40·53-s − 11.7·55-s − 6.23·59-s − 14.3·61-s + 6.87·65-s + 0.623·67-s + 13.4·71-s + 8.13·73-s + 11.7·77-s − 4.25·79-s + 15.2·83-s + ⋯ |
| L(s) = 1 | − 1.53·5-s + 1.30·7-s + 1.02·11-s − 0.556·13-s + 0.746·17-s − 0.129·19-s − 1.54·23-s + 1.35·25-s − 1.80·29-s − 0.471·31-s − 1.99·35-s − 0.0866·37-s − 0.499·41-s + 1.21·43-s + 1.01·47-s + 0.692·49-s − 0.330·53-s − 1.57·55-s − 0.812·59-s − 1.83·61-s + 0.852·65-s + 0.0761·67-s + 1.60·71-s + 0.952·73-s + 1.33·77-s − 0.479·79-s + 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 + 0.565T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 + 9.73T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 + 2.40T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 0.623T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 + 4.25T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64846916777781447360358424538, −7.43929853986200002024250593502, −6.34742492995373750662900053067, −5.46727173911445746835982763309, −4.68106041173911731099140549809, −3.96277075028458906105903591327, −3.59139854943203635400207320936, −2.21465059854270164325989712150, −1.29515050064276626236407756855, 0,
1.29515050064276626236407756855, 2.21465059854270164325989712150, 3.59139854943203635400207320936, 3.96277075028458906105903591327, 4.68106041173911731099140549809, 5.46727173911445746835982763309, 6.34742492995373750662900053067, 7.43929853986200002024250593502, 7.64846916777781447360358424538
