L(s) = 1 | + (−0.614 + 1.27i)2-s − 2.96·3-s + (−1.24 − 1.56i)4-s + (2.18 − 0.458i)5-s + (1.82 − 3.77i)6-s + i·7-s + (2.75 − 0.625i)8-s + 5.80·9-s + (−0.759 + 3.06i)10-s − 0.338i·11-s + (3.69 + 4.64i)12-s − 2.66·13-s + (−1.27 − 0.614i)14-s + (−6.49 + 1.36i)15-s + (−0.896 + 3.89i)16-s + 3.60i·17-s + ⋯ |
L(s) = 1 | + (−0.434 + 0.900i)2-s − 1.71·3-s + (−0.622 − 0.782i)4-s + (0.978 − 0.205i)5-s + (0.743 − 1.54i)6-s + 0.377i·7-s + (0.975 − 0.221i)8-s + 1.93·9-s + (−0.240 + 0.970i)10-s − 0.102i·11-s + (1.06 + 1.34i)12-s − 0.738·13-s + (−0.340 − 0.164i)14-s + (−1.67 + 0.351i)15-s + (−0.224 + 0.974i)16-s + 0.874i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.306243 + 0.477254i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.306243 + 0.477254i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.614 - 1.27i)T \) |
| 5 | \( 1 + (-2.18 + 0.458i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 11 | \( 1 + 0.338iT - 11T^{2} \) |
| 13 | \( 1 + 2.66T + 13T^{2} \) |
| 17 | \( 1 - 3.60iT - 17T^{2} \) |
| 19 | \( 1 - 7.58iT - 19T^{2} \) |
| 23 | \( 1 + 1.51iT - 23T^{2} \) |
| 29 | \( 1 - 9.39iT - 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 6.41T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 10.6iT - 47T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 + 8.53iT - 59T^{2} \) |
| 61 | \( 1 - 2.39iT - 61T^{2} \) |
| 67 | \( 1 + 6.93T + 67T^{2} \) |
| 71 | \( 1 - 0.174T + 71T^{2} \) |
| 73 | \( 1 - 9.77iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 7.09iT - 97T^{2} \) |
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
−12.35062616721762372545854453433, −10.91446256874200956128710850676, −10.25023831190835464025955266014, −9.504727186621555663750997630822, −8.226530500522859498668895325240, −6.91034766761832327563660355770, −6.02889105039224008468417444408, −5.54282615435053254326765028656, −4.54953475720372662490822783587, −1.44263015256237326867496739114,
0.67539556965927788426107811815, 2.46593420903699499569898459008, 4.49987691645577241795938459461, 5.28704891089737900785846959189, 6.58790405619663432814928284005, 7.45658450220348739143505782190, 9.254658196466469246430319243267, 9.949347294920391246133997735595, 10.68365436520970131403549545480, 11.50578403575097892706504450486