| L(s) = 1 | − 0.585·3-s − 2.65·9-s − 6.24·11-s + 5.65·17-s − 7.41·19-s − 5·25-s + 3.31·27-s + 3.65·33-s − 6·41-s − 13.0·43-s − 7·49-s − 3.31·51-s + 4.34·57-s + 14.2·59-s + 3.89·67-s + 16.9·73-s + 2.92·75-s + 6.02·81-s + 10.7·83-s − 5.65·89-s − 16.9·97-s + 16.5·99-s + 9.75·107-s − 18·113-s + ⋯ |
| L(s) = 1 | − 0.338·3-s − 0.885·9-s − 1.88·11-s + 1.37·17-s − 1.70·19-s − 25-s + 0.637·27-s + 0.636·33-s − 0.937·41-s − 1.99·43-s − 49-s − 0.464·51-s + 0.575·57-s + 1.85·59-s + 0.476·67-s + 1.98·73-s + 0.338·75-s + 0.669·81-s + 1.17·83-s − 0.599·89-s − 1.72·97-s + 1.66·99-s + 0.943·107-s − 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 7.41T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 13.0T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−10.46080621658755185286468644650, −9.817540145054609097417518595394, −8.314759993952222290464261573058, −8.054830508026680856031813388471, −6.68158415042714990375047328997, −5.63007015682856167937843064933, −5.00373285521618559565902983508, −3.45139653008737306422731937141, −2.27097777800179806952192006462, 0,
2.27097777800179806952192006462, 3.45139653008737306422731937141, 5.00373285521618559565902983508, 5.63007015682856167937843064933, 6.68158415042714990375047328997, 8.054830508026680856031813388471, 8.314759993952222290464261573058, 9.817540145054609097417518595394, 10.46080621658755185286468644650
