| L(s) = 1 | + 3.04·5-s − 7-s + 6.09·11-s − 13-s + 6.92·17-s + 5.27·19-s + 3.04·23-s + 4.27·25-s − 3.88·29-s − 1.27·31-s − 3.04·35-s − 4.54·37-s − 6.92·41-s − 11.8·43-s − 3.04·47-s + 49-s + 2.20·53-s + 18.5·55-s + 0.837·59-s − 2·61-s − 3.04·65-s + 14.5·67-s − 12.1·71-s + 7.27·73-s − 6.09·77-s + 13.2·79-s + 9.13·83-s + ⋯ |
| L(s) = 1 | + 1.36·5-s − 0.377·7-s + 1.83·11-s − 0.277·13-s + 1.68·17-s + 1.21·19-s + 0.635·23-s + 0.854·25-s − 0.721·29-s − 0.228·31-s − 0.514·35-s − 0.747·37-s − 1.08·41-s − 1.80·43-s − 0.444·47-s + 0.142·49-s + 0.303·53-s + 2.50·55-s + 0.109·59-s − 0.256·61-s − 0.377·65-s + 1.77·67-s − 1.44·71-s + 0.851·73-s − 0.694·77-s + 1.49·79-s + 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.866625854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.866625854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 - 2.20T + 53T^{2} \) |
| 59 | \( 1 - 0.837T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 7.27T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 9.13T + 83T^{2} \) |
| 89 | \( 1 - 4.71T + 89T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−8.896206959106589279444968214510, −7.88418076016473034264413994105, −6.88290940220872256712836616823, −6.50286789394292603708635715795, −5.51454485261788701722177479625, −5.14412325674495307075185079290, −3.71050308853981573368640231738, −3.17424157587407430261574866121, −1.82584152525825290536030124278, −1.14368846265696305706877237949,
1.14368846265696305706877237949, 1.82584152525825290536030124278, 3.17424157587407430261574866121, 3.71050308853981573368640231738, 5.14412325674495307075185079290, 5.51454485261788701722177479625, 6.50286789394292603708635715795, 6.88290940220872256712836616823, 7.88418076016473034264413994105, 8.896206959106589279444968214510
