| L(s) = 1 | − 2.47·2-s + 3-s + 4.11·4-s − 2.58·5-s − 2.47·6-s + 7-s − 5.22·8-s + 9-s + 6.39·10-s + 4.11·12-s + 5.87·13-s − 2.47·14-s − 2.58·15-s + 4.70·16-s − 7.51·17-s − 2.47·18-s + 2.35·19-s − 10.6·20-s + 21-s + 6.94·23-s − 5.22·24-s + 1.69·25-s − 14.5·26-s + 27-s + 4.11·28-s + 5.87·29-s + 6.39·30-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.577·3-s + 2.05·4-s − 1.15·5-s − 1.00·6-s + 0.377·7-s − 1.84·8-s + 0.333·9-s + 2.02·10-s + 1.18·12-s + 1.62·13-s − 0.660·14-s − 0.668·15-s + 1.17·16-s − 1.82·17-s − 0.582·18-s + 0.540·19-s − 2.38·20-s + 0.218·21-s + 1.44·23-s − 1.06·24-s + 0.339·25-s − 2.84·26-s + 0.192·27-s + 0.777·28-s + 1.09·29-s + 1.16·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8010228940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8010228940\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 + 7.40T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 + 4.22T + 53T^{2} \) |
| 59 | \( 1 + 0.926T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.45T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.922003803359020506271078343838, −8.258972688522105520500741637295, −7.74683038064807129637161508953, −6.92566815954990692739202678962, −6.38776070074322086517278022532, −4.83956805675981488830568088611, −3.84627894650337161500993585141, −2.92866045064242928514214694633, −1.76768107474513905905470151994, −0.73468646983121734777352245572,
0.73468646983121734777352245572, 1.76768107474513905905470151994, 2.92866045064242928514214694633, 3.84627894650337161500993585141, 4.83956805675981488830568088611, 6.38776070074322086517278022532, 6.92566815954990692739202678962, 7.74683038064807129637161508953, 8.258972688522105520500741637295, 8.922003803359020506271078343838
