L(s) = 1 | + 0.760·2-s − 1.42·4-s − 3.18·5-s − 2.60·8-s − 2.42·10-s + 2.23·11-s − 3.70·13-s + 0.861·16-s + 5.60·17-s − 4.42·19-s + 4.52·20-s + 1.70·22-s − 0.942·23-s + 5.12·25-s − 2.81·26-s + 10.1·29-s + 5.70·31-s + 5.86·32-s + 4.26·34-s + 3.12·37-s − 3.36·38-s + 8.28·40-s − 3.98·41-s − 3.28·43-s − 3.18·44-s − 0.717·46-s + 0.225·47-s + ⋯ |
L(s) = 1 | + 0.538·2-s − 0.710·4-s − 1.42·5-s − 0.920·8-s − 0.765·10-s + 0.675·11-s − 1.02·13-s + 0.215·16-s + 1.35·17-s − 1.01·19-s + 1.01·20-s + 0.363·22-s − 0.196·23-s + 1.02·25-s − 0.552·26-s + 1.88·29-s + 1.02·31-s + 1.03·32-s + 0.731·34-s + 0.513·37-s − 0.545·38-s + 1.30·40-s − 0.622·41-s − 0.500·43-s − 0.479·44-s − 0.105·46-s + 0.0328·47-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
\(L(1)\) |
\(\approx\) |
\(1.132077402\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132077402\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.760T + 2T^{2} \) |
| 5 | \( 1 + 3.18T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 + 4.42T + 19T^{2} \) |
| 23 | \( 1 + 0.942T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 3.98T + 41T^{2} \) |
| 43 | \( 1 + 3.28T + 43T^{2} \) |
| 47 | \( 1 - 0.225T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 5.84T + 61T^{2} \) |
| 67 | \( 1 - 7.42T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + 7.55T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 - 8.11T + 83T^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 + 0.842T + 97T^{2} \) |
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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
−9.727285742764224034032361197842, −8.510972566094292981428204874845, −8.221145893715625897739793843526, −7.19618995858780359176017427665, −6.30451642111038210770496949945, −5.14364003819429536585102039665, −4.40694138984125799620544291856, −3.76575394111948690607787978118, −2.80627788180869429299571593262, −0.71654060367547032107368546326,
0.71654060367547032107368546326, 2.80627788180869429299571593262, 3.76575394111948690607787978118, 4.40694138984125799620544291856, 5.14364003819429536585102039665, 6.30451642111038210770496949945, 7.19618995858780359176017427665, 8.221145893715625897739793843526, 8.510972566094292981428204874845, 9.727285742764224034032361197842