| L(s) = 1 | + 0.770·3-s + 3.57·5-s − 0.727·7-s − 2.40·9-s − 0.908·11-s − 4.34·13-s + 2.75·15-s − 1.93·17-s − 4.49·19-s − 0.560·21-s − 2.85·23-s + 7.77·25-s − 4.16·27-s + 2.23·29-s − 7.09·31-s − 0.700·33-s − 2.60·35-s + 7.50·37-s − 3.35·39-s + 6.45·41-s + 4.77·43-s − 8.60·45-s − 11.4·47-s − 6.47·49-s − 1.48·51-s − 6.08·53-s − 3.24·55-s + ⋯ |
| L(s) = 1 | + 0.444·3-s + 1.59·5-s − 0.274·7-s − 0.801·9-s − 0.273·11-s − 1.20·13-s + 0.711·15-s − 0.468·17-s − 1.03·19-s − 0.122·21-s − 0.594·23-s + 1.55·25-s − 0.801·27-s + 0.415·29-s − 1.27·31-s − 0.121·33-s − 0.439·35-s + 1.23·37-s − 0.536·39-s + 1.00·41-s + 0.728·43-s − 1.28·45-s − 1.67·47-s − 0.924·49-s − 0.208·51-s − 0.836·53-s − 0.438·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.770T + 3T^{2} \) |
| 5 | \( 1 - 3.57T + 5T^{2} \) |
| 7 | \( 1 + 0.727T + 7T^{2} \) |
| 11 | \( 1 + 0.908T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 + 4.49T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 + 7.09T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 + 2.77T + 59T^{2} \) |
| 61 | \( 1 + 3.49T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 + 3.61T + 73T^{2} \) |
| 79 | \( 1 - 3.39T + 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.143084243543917149296232088661, −7.40018082457092688647190019642, −6.33930021616543065135668362225, −6.02150920234793356598879256517, −5.16035362108674850452848992755, −4.38466196200192307545153216603, −3.06928644590553552026111129050, −2.42749690499505018682263033789, −1.80045252670244753853141659915, 0,
1.80045252670244753853141659915, 2.42749690499505018682263033789, 3.06928644590553552026111129050, 4.38466196200192307545153216603, 5.16035362108674850452848992755, 6.02150920234793356598879256517, 6.33930021616543065135668362225, 7.40018082457092688647190019642, 8.143084243543917149296232088661
