| L(s) = 1 | + 2.28·2-s + 3.23·4-s + 4.23·5-s + 2.82·8-s + 9.69·10-s − 3.70·11-s − 2.28·13-s + 5·17-s + 5.45·19-s + 13.7·20-s − 8.47·22-s + 0.333·23-s + 12.9·25-s − 5.23·26-s − 7.19·29-s − 3.36·31-s − 5.65·32-s + 11.4·34-s − 4.70·37-s + 12.4·38-s + 11.9·40-s − 4.70·41-s + 2.23·43-s − 11.9·44-s + 0.763·46-s − 1.47·47-s + 29.6·50-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.61·4-s + 1.89·5-s + 0.999·8-s + 3.06·10-s − 1.11·11-s − 0.634·13-s + 1.21·17-s + 1.25·19-s + 3.06·20-s − 1.80·22-s + 0.0696·23-s + 2.58·25-s − 1.02·26-s − 1.33·29-s − 0.605·31-s − 1.00·32-s + 1.96·34-s − 0.774·37-s + 2.02·38-s + 1.89·40-s − 0.735·41-s + 0.340·43-s − 1.80·44-s + 0.112·46-s − 0.214·47-s + 4.18·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.297421678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.297421678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.28T + 2T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 - 0.333T + 23T^{2} \) |
| 29 | \( 1 + 7.19T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.746T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 0.236T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 1.95T + 97T^{2} \) |
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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
−9.830446269494373846369680484278, −9.058586561031780100693465929271, −7.62730013979029889669754912775, −6.88769986878367372276323841186, −5.76352478425873227598209271367, −5.47025987925842309070999962783, −4.91857185432152356013645186868, −3.41330894915159412570997971449, −2.65461827261119141779263812211, −1.72287427867671709484176966283,
1.72287427867671709484176966283, 2.65461827261119141779263812211, 3.41330894915159412570997971449, 4.91857185432152356013645186868, 5.47025987925842309070999962783, 5.76352478425873227598209271367, 6.88769986878367372276323841186, 7.62730013979029889669754912775, 9.058586561031780100693465929271, 9.830446269494373846369680484278
