| L(s) = 1 | + 3-s + 3.61·5-s − 2.72·7-s + 9-s − 4.28·11-s − 0.0474·13-s + 3.61·15-s + 3.70·17-s − 0.502·19-s − 2.72·21-s − 2.97·23-s + 8.08·25-s + 27-s − 6.25·29-s − 5.93·31-s − 4.28·33-s − 9.86·35-s + 0.158·37-s − 0.0474·39-s − 3.89·41-s − 4.94·43-s + 3.61·45-s − 8.39·47-s + 0.440·49-s + 3.70·51-s − 10.7·53-s − 15.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.61·5-s − 1.03·7-s + 0.333·9-s − 1.29·11-s − 0.0131·13-s + 0.933·15-s + 0.899·17-s − 0.115·19-s − 0.595·21-s − 0.621·23-s + 1.61·25-s + 0.192·27-s − 1.16·29-s − 1.06·31-s − 0.745·33-s − 1.66·35-s + 0.0260·37-s − 0.00759·39-s − 0.608·41-s − 0.754·43-s + 0.539·45-s − 1.22·47-s + 0.0629·49-s + 0.519·51-s − 1.47·53-s − 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 2.72T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 + 0.0474T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 0.502T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 + 5.93T + 31T^{2} \) |
| 37 | \( 1 - 0.158T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.54T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 + 0.477T + 67T^{2} \) |
| 71 | \( 1 + 0.963T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 1.53T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 + 1.19T + 97T^{2} \) |
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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
−7.52968839904298485990094843645, −6.71329684970582607989592406928, −6.05992753285001037956459271247, −5.48576941860191337747791706975, −4.88823934535730197050297189554, −3.58037185492773101395328845762, −3.06234103453882132118329169449, −2.22164552089705022849597504044, −1.58933089237121421702711929855, 0,
1.58933089237121421702711929855, 2.22164552089705022849597504044, 3.06234103453882132118329169449, 3.58037185492773101395328845762, 4.88823934535730197050297189554, 5.48576941860191337747791706975, 6.05992753285001037956459271247, 6.71329684970582607989592406928, 7.52968839904298485990094843645
