L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (2.23 + 0.0994i)5-s + i·6-s + (2.52 + 0.781i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.482 − 2.18i)10-s + (−1.31 − 2.27i)11-s + (0.965 − 0.258i)12-s + (1.21 − 1.21i)13-s + (0.101 − 2.64i)14-s + (−2.13 − 0.674i)15-s + (0.500 − 0.866i)16-s + (1.95 − 7.31i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.999 + 0.0444i)5-s + 0.408i·6-s + (0.955 + 0.295i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.152 − 0.690i)10-s + (−0.395 − 0.685i)11-s + (0.278 − 0.0747i)12-s + (0.337 − 0.337i)13-s + (0.0270 − 0.706i)14-s + (−0.550 − 0.174i)15-s + (0.125 − 0.216i)16-s + (0.475 − 1.77i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.979727 - 0.551143i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.979727 - 0.551143i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (-2.23 - 0.0994i)T \) |
| 7 | \( 1 + (-2.52 - 0.781i)T \) |
good | 11 | \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 1.21i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.95 + 7.31i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.32 - 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.95 + 1.32i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.99iT - 29T^{2} \) |
| 31 | \( 1 + (8.66 - 5.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.02 - 3.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.59iT - 41T^{2} \) |
| 43 | \( 1 + (0.545 + 0.545i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.12 - 1.64i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.22 - 8.28i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.86 + 6.68i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.16 - 2.40i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.47 - 0.663i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 + (13.1 + 3.53i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.78 - 4.49i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.99 - 7.99i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.0812 - 0.140i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.35 + 4.35i)T + 97iT^{2} \) |
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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
−12.14537350416867862583505085442, −11.04507956498584094916462654109, −10.59238070501938192477663429303, −9.364733857447104337769082233826, −8.453240913900027045088478759885, −7.14764267298488118297802564895, −5.62998428648627567478449091595, −4.98562345955490397130116379374, −2.97526938828148939816596502655, −1.42156759447870718443304892647,
1.73337544208187731407185567743, 4.29900762805221161823569372309, 5.33538223472321635525484536092, 6.25815759495283108677232834022, 7.37962779944948299140920155652, 8.524989026848482893918725889969, 9.598363437562670933846510062925, 10.53593888125773620686517771014, 11.30031008531124999883664875591, 12.85434444283291852224403409803