| L(s) = 1 | + 2.62·3-s − 0.385·5-s + 1.06·7-s + 3.91·9-s − 3.06·11-s − 4.43·13-s − 1.01·15-s − 2.91·17-s − 0.422·19-s + 2.78·21-s − 0.921·23-s − 4.85·25-s + 2.39·27-s − 8.01·29-s + 6.77·31-s − 8.06·33-s − 0.409·35-s + 4.94·37-s − 11.6·39-s + 7.67·41-s + 1.55·43-s − 1.50·45-s − 4.43·47-s − 5.87·49-s − 7.67·51-s − 9.11·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.172·5-s + 0.400·7-s + 1.30·9-s − 0.924·11-s − 1.23·13-s − 0.261·15-s − 0.707·17-s − 0.0968·19-s + 0.608·21-s − 0.192·23-s − 0.970·25-s + 0.460·27-s − 1.48·29-s + 1.21·31-s − 1.40·33-s − 0.0691·35-s + 0.812·37-s − 1.86·39-s + 1.19·41-s + 0.237·43-s − 0.225·45-s − 0.647·47-s − 0.839·49-s − 1.07·51-s − 1.25·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 + 0.385T + 5T^{2} \) |
| 7 | \( 1 - 1.06T + 7T^{2} \) |
| 11 | \( 1 + 3.06T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 + 0.422T + 19T^{2} \) |
| 23 | \( 1 + 0.921T + 23T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.11T + 89T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−7.76205406943421812073404127945, −7.48849257954793272950671112342, −6.44244616788502112976584449207, −5.46970094795678525658720282526, −4.59479367704574985513795090350, −4.04349347496327145165085174565, −2.98794237175640436094389672849, −2.45767573965225957352217390402, −1.73437748517393853595401009380, 0,
1.73437748517393853595401009380, 2.45767573965225957352217390402, 2.98794237175640436094389672849, 4.04349347496327145165085174565, 4.59479367704574985513795090350, 5.46970094795678525658720282526, 6.44244616788502112976584449207, 7.48849257954793272950671112342, 7.76205406943421812073404127945
