| L(s) = 1 | + 2.36·2-s − 3-s + 3.57·4-s − 3.93·5-s − 2.36·6-s + 7-s + 3.72·8-s + 9-s − 9.29·10-s − 3.57·12-s + 3.93·13-s + 2.36·14-s + 3.93·15-s + 1.63·16-s − 4.72·17-s + 2.36·18-s − 4.78·19-s − 14.0·20-s − 21-s + 2.72·23-s − 3.72·24-s + 10.5·25-s + 9.29·26-s − 27-s + 3.57·28-s − 7.93·29-s + 9.29·30-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.78·4-s − 1.76·5-s − 0.964·6-s + 0.377·7-s + 1.31·8-s + 0.333·9-s − 2.94·10-s − 1.03·12-s + 1.09·13-s + 0.631·14-s + 1.01·15-s + 0.409·16-s − 1.14·17-s + 0.556·18-s − 1.09·19-s − 3.14·20-s − 0.218·21-s + 0.567·23-s − 0.759·24-s + 2.10·25-s + 1.82·26-s − 0.192·27-s + 0.675·28-s − 1.47·29-s + 1.69·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 5 | \( 1 + 3.93T + 5T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 + 4.72T + 17T^{2} \) |
| 19 | \( 1 + 4.78T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 1.15T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 + 0.430T + 41T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 + 8.78T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 - 2.84T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.320500802153162083722408044378, −7.48488140191588829186346028970, −6.69813461076798671968131709079, −6.18293208556781631957851554631, −5.02446731279496458372469275063, −4.55268319661071526485654652611, −3.82639202028721170137771871329, −3.26089788392685103243791434699, −1.81462131850239465988044417699, 0,
1.81462131850239465988044417699, 3.26089788392685103243791434699, 3.82639202028721170137771871329, 4.55268319661071526485654652611, 5.02446731279496458372469275063, 6.18293208556781631957851554631, 6.69813461076798671968131709079, 7.48488140191588829186346028970, 8.320500802153162083722408044378
