| L(s) = 1 | + 3-s − 0.922·5-s + 7-s + 9-s + 0.382·11-s − 4.13·13-s − 0.922·15-s − 7.05·17-s − 1.30·19-s + 21-s − 2.44·23-s − 4.14·25-s + 27-s + 5.30·29-s + 10.9·31-s + 0.382·33-s − 0.922·35-s + 10.1·37-s − 4.13·39-s − 4.44·41-s − 0.983·43-s − 0.922·45-s + 7.50·47-s + 49-s − 7.05·51-s + 7.14·53-s − 0.352·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.412·5-s + 0.377·7-s + 0.333·9-s + 0.115·11-s − 1.14·13-s − 0.238·15-s − 1.71·17-s − 0.299·19-s + 0.218·21-s − 0.510·23-s − 0.829·25-s + 0.192·27-s + 0.985·29-s + 1.96·31-s + 0.0665·33-s − 0.155·35-s + 1.67·37-s − 0.661·39-s − 0.694·41-s − 0.150·43-s − 0.137·45-s + 1.09·47-s + 0.142·49-s − 0.987·51-s + 0.982·53-s − 0.0475·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.985670496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.985670496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 0.922T + 5T^{2} \) |
| 11 | \( 1 - 0.382T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 + 1.30T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 + 0.983T + 43T^{2} \) |
| 47 | \( 1 - 7.50T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 - 6.51T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 - 9.50T + 73T^{2} \) |
| 79 | \( 1 - 3.81T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−8.247005579688466412410870419581, −7.57441147352036106096565269702, −6.80805189428463985011461315822, −6.21711672567293166425383732736, −5.03132343017723624145589833957, −4.41991128955583620286566928089, −3.86093293540110797451626506464, −2.53199765661831315403738181698, −2.24644370940313937654102794808, −0.71635797505961202000738950155,
0.71635797505961202000738950155, 2.24644370940313937654102794808, 2.53199765661831315403738181698, 3.86093293540110797451626506464, 4.41991128955583620286566928089, 5.03132343017723624145589833957, 6.21711672567293166425383732736, 6.80805189428463985011461315822, 7.57441147352036106096565269702, 8.247005579688466412410870419581
