| L(s) = 1 | − 3.30·3-s + 0.302·5-s + 7-s + 7.90·9-s − 2.60·11-s + 0.605·13-s − 1.00·15-s − 17-s − 6·19-s − 3.30·21-s + 4.60·23-s − 4.90·25-s − 16.2·27-s − 1.39·29-s − 10.3·31-s + 8.60·33-s + 0.302·35-s − 7.21·37-s − 2.00·39-s − 8.51·41-s − 0.697·43-s + 2.39·45-s + 10·47-s + 49-s + 3.30·51-s + 4.30·53-s − 0.788·55-s + ⋯ |
| L(s) = 1 | − 1.90·3-s + 0.135·5-s + 0.377·7-s + 2.63·9-s − 0.785·11-s + 0.167·13-s − 0.258·15-s − 0.242·17-s − 1.37·19-s − 0.720·21-s + 0.960·23-s − 0.981·25-s − 3.11·27-s − 0.258·29-s − 1.85·31-s + 1.49·33-s + 0.0511·35-s − 1.18·37-s − 0.320·39-s − 1.32·41-s − 0.106·43-s + 0.356·45-s + 1.45·47-s + 0.142·49-s + 0.462·51-s + 0.591·53-s − 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 - 0.302T + 5T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.21T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 - 10T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 0.697T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 5.21T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−10.79420332340129358871089145272, −10.08964048887404750722789904321, −8.838366900915876296429655943828, −7.50073954977690814783436592085, −6.70385753197726502977657401306, −5.69114875933762450620973709082, −5.09277283063707884389548729149, −4.01621350754607537551052472636, −1.80826892973584175146586308601, 0,
1.80826892973584175146586308601, 4.01621350754607537551052472636, 5.09277283063707884389548729149, 5.69114875933762450620973709082, 6.70385753197726502977657401306, 7.50073954977690814783436592085, 8.838366900915876296429655943828, 10.08964048887404750722789904321, 10.79420332340129358871089145272
