| L(s) = 1 | + 2.22·2-s + 3-s + 2.96·4-s + 2.15·5-s + 2.22·6-s + 2.15·8-s + 9-s + 4.79·10-s − 11-s + 2.96·12-s + 2.62·13-s + 2.15·15-s − 1.13·16-s + 1.89·17-s + 2.22·18-s − 1.45·19-s + 6.37·20-s − 2.22·22-s − 2.19·23-s + 2.15·24-s − 0.371·25-s + 5.85·26-s + 27-s + 3.32·29-s + 4.79·30-s + 9.02·31-s − 6.83·32-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 0.577·3-s + 1.48·4-s + 0.962·5-s + 0.909·6-s + 0.760·8-s + 0.333·9-s + 1.51·10-s − 0.301·11-s + 0.856·12-s + 0.728·13-s + 0.555·15-s − 0.284·16-s + 0.458·17-s + 0.525·18-s − 0.334·19-s + 1.42·20-s − 0.475·22-s − 0.457·23-s + 0.439·24-s − 0.0743·25-s + 1.14·26-s + 0.192·27-s + 0.617·29-s + 0.875·30-s + 1.62·31-s − 1.20·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.817159836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.817159836\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 - 2.15T + 5T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 9.02T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 + 0.516T + 41T^{2} \) |
| 43 | \( 1 + 1.30T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 9.30T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.453234274985086700360329472954, −8.565463909494172713010815997658, −7.68798757325468934149145980100, −6.50018884871016235869343391222, −6.11346857559922304897076885356, −5.19288007345571420627861091648, −4.40391162659744849968947924430, −3.42867583371439610035841634834, −2.65022077922472056393375089215, −1.65764887447277033798155044987,
1.65764887447277033798155044987, 2.65022077922472056393375089215, 3.42867583371439610035841634834, 4.40391162659744849968947924430, 5.19288007345571420627861091648, 6.11346857559922304897076885356, 6.50018884871016235869343391222, 7.68798757325468934149145980100, 8.565463909494172713010815997658, 9.453234274985086700360329472954
