L(s) = 1 | + (−0.309 − 0.951i)2-s + (−1.02 + 3.15i)3-s + (−0.809 + 0.587i)4-s + 2.66·5-s + 3.31·6-s + (−0.5 + 0.363i)7-s + (0.809 + 0.587i)8-s + (−6.45 − 4.69i)9-s + (−0.823 − 2.53i)10-s + (1.09 − 0.794i)11-s + (−1.02 − 3.15i)12-s + (0.729 − 2.24i)13-s + (0.5 + 0.363i)14-s + (−2.72 + 8.40i)15-s + (0.309 − 0.951i)16-s + (2.15 + 1.56i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.591 + 1.81i)3-s + (−0.404 + 0.293i)4-s + 1.19·5-s + 1.35·6-s + (−0.188 + 0.137i)7-s + (0.286 + 0.207i)8-s + (−2.15 − 1.56i)9-s + (−0.260 − 0.801i)10-s + (0.329 − 0.239i)11-s + (−0.295 − 0.909i)12-s + (0.202 − 0.622i)13-s + (0.133 + 0.0970i)14-s + (−0.704 + 2.16i)15-s + (0.0772 − 0.237i)16-s + (0.523 + 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.703652 + 0.246866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.703652 + 0.246866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (5.42 + 1.26i)T \) |
good | 3 | \( 1 + (1.02 - 3.15i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 + (0.5 - 0.363i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.09 + 0.794i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.729 + 2.24i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.15 - 1.56i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.98 + 6.09i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.37 - 1.72i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.17 - 3.62i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + 4.57T + 37T^{2} \) |
| 41 | \( 1 + (0.676 + 2.08i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.92 - 5.93i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (1.5 - 4.61i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.78 + 2.02i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.42 + 10.5i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 + 9.40T + 67T^{2} \) |
| 71 | \( 1 + (6.62 + 4.81i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.47 + 2.52i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.64 - 1.91i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.354 + 1.08i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (6.85 - 4.98i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.42 + 1.76i)T + (29.9 - 92.2i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.23612411586198716042624535421, −14.15136584193962658896336350922, −12.77340118398644771838113325149, −11.25935508714175524717406010206, −10.54231100778111091215952717236, −9.580925548944279506656268402387, −8.906805851525785977659568773206, −6.02729670419625671369950665219, −4.92051164826359304092076805227, −3.25657413500718852002690975402,
1.74554332547384636926130650134, 5.57341494907444879961472610434, 6.40721647464635013961024553604, 7.36600666878995911979968219896, 8.742192915670178633708992873828, 10.26657057317951294558754532546, 11.85059406007025000204052552386, 12.89366807116952255256232287314, 13.77252495240211716704911088058, 14.45961851026902772115026016041