| L(s) = 1 | + 2.77·2-s − 1.12·3-s + 5.69·4-s − 2.49·5-s − 3.12·6-s + 10.2·8-s − 1.72·9-s − 6.92·10-s + 11-s − 6.41·12-s + 13-s + 2.81·15-s + 17.0·16-s − 0.315·17-s − 4.79·18-s + 1.38·19-s − 14.2·20-s + 2.77·22-s + 3.22·23-s − 11.5·24-s + 1.23·25-s + 2.77·26-s + 5.33·27-s − 0.210·29-s + 7.80·30-s − 2.20·31-s + 26.7·32-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 0.650·3-s + 2.84·4-s − 1.11·5-s − 1.27·6-s + 3.62·8-s − 0.576·9-s − 2.19·10-s + 0.301·11-s − 1.85·12-s + 0.277·13-s + 0.726·15-s + 4.25·16-s − 0.0764·17-s − 1.13·18-s + 0.317·19-s − 3.18·20-s + 0.591·22-s + 0.671·23-s − 2.35·24-s + 0.247·25-s + 0.543·26-s + 1.02·27-s − 0.0391·29-s + 1.42·30-s − 0.395·31-s + 4.73·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.176289287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.176289287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 17 | \( 1 + 0.315T + 17T^{2} \) |
| 19 | \( 1 - 1.38T + 19T^{2} \) |
| 23 | \( 1 - 3.22T + 23T^{2} \) |
| 29 | \( 1 + 0.210T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 8.95T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 0.337T + 59T^{2} \) |
| 61 | \( 1 + 0.896T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 3.09T + 71T^{2} \) |
| 73 | \( 1 + 0.985T + 73T^{2} \) |
| 79 | \( 1 - 9.30T + 79T^{2} \) |
| 83 | \( 1 - 6.45T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.68180451065153205766982817336, −6.90158628959746974106136823216, −6.35885682610829311305151782959, −5.69104501672944950531764957486, −5.02758407687008152801211925214, −4.43724691902463341645766242463, −3.66906039223906774988113220251, −3.16477148337398063887150291739, −2.21156141701529799617086316299, −0.893407237410525425991472507802,
0.893407237410525425991472507802, 2.21156141701529799617086316299, 3.16477148337398063887150291739, 3.66906039223906774988113220251, 4.43724691902463341645766242463, 5.02758407687008152801211925214, 5.69104501672944950531764957486, 6.35885682610829311305151782959, 6.90158628959746974106136823216, 7.68180451065153205766982817336
