L(s) = 1 | + (1.16 + 1.16i)5-s + 3.74·7-s + (1.64 − 1.64i)11-s + (0.645 + 0.645i)13-s + 6.57i·17-s + (1.41 − 1.41i)19-s + 6i·23-s − 2.29i·25-s + (5.40 − 5.40i)29-s + 0.913i·31-s + (4.35 + 4.35i)35-s + (−1 + i)37-s − 6.57·41-s + (3.74 + 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (0.520 + 0.520i)5-s + 1.41·7-s + (0.496 − 0.496i)11-s + (0.179 + 0.179i)13-s + 1.59i·17-s + (0.324 − 0.324i)19-s + 1.25i·23-s − 0.458i·25-s + (1.00 − 1.00i)29-s + 0.164i·31-s + (0.736 + 0.736i)35-s + (−0.164 + 0.164i)37-s − 1.02·41-s + (0.570 + 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.512465257\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.512465257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.16 - 1.16i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.645 - 0.645i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.57iT - 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-5.40 + 5.40i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.913iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.57T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (7.73 + 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (9.29 - 9.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (10.2 + 10.2i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.70iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 + (-10.9 - 10.9i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.412T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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
−9.005685766333566072743635585583, −8.194476376115388877208887202900, −7.75715949889073427754364660573, −6.56467720340369688725007410140, −6.06175977267467158906727440681, −5.12712566368821917191512970208, −4.26318022291093245411848435364, −3.32301647953333059968779938193, −2.08363783911096590762084080864, −1.30067307837659604835628955409,
1.01706857206708345040071740210, 1.89380228257844443205358829284, 3.01172859064016368825186161672, 4.40445755542395875933978951930, 4.91228134632367131012318970314, 5.58482129336705470878315318569, 6.69318757213752415439060930736, 7.44283862673824548234434381740, 8.220537996615994210274523589314, 8.992331115227246037051488211258