| L(s) = 1 | − 2.64i·2-s + i·3-s − 5.00·4-s + 2.64·6-s + 1.64i·7-s + 7.93i·8-s − 9-s + 0.354·11-s − 5.00i·12-s + 0.354i·13-s + 4.35·14-s + 11.0·16-s − 4i·17-s + 2.64i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 1.87i·2-s + 0.577i·3-s − 2.50·4-s + 1.08·6-s + 0.622i·7-s + 2.80i·8-s − 0.333·9-s + 0.106·11-s − 1.44i·12-s + 0.0982i·13-s + 1.16·14-s + 2.75·16-s − 0.970i·17-s + 0.623i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7535482456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7535482456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.64iT - 2T^{2} \) |
| 7 | \( 1 - 1.64iT - 7T^{2} \) |
| 11 | \( 1 - 0.354T + 11T^{2} \) |
| 13 | \( 1 - 0.354iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 + 9.29iT - 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 3.64iT - 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 5.64iT - 43T^{2} \) |
| 47 | \( 1 + 1.29iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 6.58iT - 67T^{2} \) |
| 71 | \( 1 - 7.29T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 16.9T + 89T^{2} \) |
| 97 | \( 1 + 2.93iT - 97T^{2} \) |
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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
−9.180381506936532248878288031303, −8.892487044417369655234754916948, −7.909254499103621116219769804310, −6.42035845323397024182214404453, −5.17008533026142241298646497908, −4.66642961651913353231932962902, −3.61954067693656378523645027394, −2.81803512599027198934523148644, −1.93039957994342878797821849001, −0.32673721334488739356012600522,
1.35917482622367926127006304528, 3.46530521443920727581844780315, 4.33575912099777727312945573373, 5.43909063193821190254668619828, 6.00884736493140656197035006263, 6.84659128051662348087991197991, 7.60552874571891845520799414961, 7.920562302100694830952547002371, 8.954773302630430381247419669096, 9.566197283392287097447848998737
