| L(s) = 1 | − 2.76·2-s − 2.85·3-s + 5.66·4-s − 3.02·5-s + 7.89·6-s − 10.1·8-s + 5.12·9-s + 8.36·10-s − 0.809·11-s − 16.1·12-s − 3.86·13-s + 8.61·15-s + 16.7·16-s − 0.828·17-s − 14.1·18-s − 1.89·19-s − 17.1·20-s + 2.24·22-s − 0.475·23-s + 28.8·24-s + 4.12·25-s + 10.7·26-s − 6.05·27-s − 2.61·29-s − 23.8·30-s + 0.449·31-s − 26.0·32-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.64·3-s + 2.83·4-s − 1.35·5-s + 3.22·6-s − 3.58·8-s + 1.70·9-s + 2.64·10-s − 0.244·11-s − 4.65·12-s − 1.07·13-s + 2.22·15-s + 4.18·16-s − 0.200·17-s − 3.34·18-s − 0.434·19-s − 3.82·20-s + 0.477·22-s − 0.0991·23-s + 5.89·24-s + 0.825·25-s + 2.09·26-s − 1.16·27-s − 0.484·29-s − 4.35·30-s + 0.0806·31-s − 4.60·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02521913990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02521913990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 2.85T + 3T^{2} \) |
| 5 | \( 1 + 3.02T + 5T^{2} \) |
| 11 | \( 1 + 0.809T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 + 0.475T + 23T^{2} \) |
| 29 | \( 1 + 2.61T + 29T^{2} \) |
| 31 | \( 1 - 0.449T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 43 | \( 1 + 6.10T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 9.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 + 1.45T + 71T^{2} \) |
| 73 | \( 1 - 8.74T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 + 6.30T + 83T^{2} \) |
| 89 | \( 1 + 3.07T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−9.322970897725299480621659094352, −8.137224416276045952387259855164, −7.75860252622971089073483949635, −6.93741183196709617215495632726, −6.43756264999438106016166386983, −5.41405783971273291586609753946, −4.36424618962937421035243739225, −2.94772950115792795270845237121, −1.52811704052336574364848209037, −0.15504788693127429845526275072,
0.15504788693127429845526275072, 1.52811704052336574364848209037, 2.94772950115792795270845237121, 4.36424618962937421035243739225, 5.41405783971273291586609753946, 6.43756264999438106016166386983, 6.93741183196709617215495632726, 7.75860252622971089073483949635, 8.137224416276045952387259855164, 9.322970897725299480621659094352
