L(s) = 1 | + (1.55 + 1.26i)2-s + (0.815 + 3.91i)4-s − 2.61·5-s − 10.9i·7-s + (−3.67 + 7.10i)8-s + (−4.05 − 3.29i)10-s + 21.8i·11-s − 4.95·13-s + (13.8 − 17.0i)14-s + (−14.6 + 6.38i)16-s − 20.1·17-s + 4.35i·19-s + (−2.13 − 10.2i)20-s + (−27.5 + 33.9i)22-s − 9.15i·23-s + ⋯ |
L(s) = 1 | + (0.775 + 0.630i)2-s + (0.203 + 0.978i)4-s − 0.522·5-s − 1.57i·7-s + (−0.459 + 0.888i)8-s + (−0.405 − 0.329i)10-s + 1.98i·11-s − 0.381·13-s + (0.990 − 1.21i)14-s + (−0.916 + 0.399i)16-s − 1.18·17-s + 0.229i·19-s + (−0.106 − 0.511i)20-s + (−1.25 + 1.54i)22-s − 0.398i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9210847138\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9210847138\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.55 - 1.26i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 + 2.61T + 25T^{2} \) |
| 7 | \( 1 + 10.9iT - 49T^{2} \) |
| 11 | \( 1 - 21.8iT - 121T^{2} \) |
| 13 | \( 1 + 4.95T + 169T^{2} \) |
| 17 | \( 1 + 20.1T + 289T^{2} \) |
| 23 | \( 1 + 9.15iT - 529T^{2} \) |
| 29 | \( 1 + 1.24T + 841T^{2} \) |
| 31 | \( 1 - 39.9iT - 961T^{2} \) |
| 37 | \( 1 + 56.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 31.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 82.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 16.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 29.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 21.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 74.7T + 9.40e3T^{2} \) |
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
−10.71812841990172569721970864648, −10.03148507510032427033523729370, −8.806201134740277951748417404718, −7.68809723480896924483537642854, −7.14241347577473009174347535179, −6.63670094786929602142268157689, −4.97559902842683861347743093090, −4.39430650901955520427786855510, −3.62760578733177185604801637724, −2.02162818321676907018965966874,
0.23238386788116055232568005388, 2.11559388817479317435766233279, 3.06306415776007352863277800659, 4.04407994678327005945213593587, 5.39174484690835141468047761798, 5.82811078510175699636707545129, 6.89528323967679613894552774657, 8.437896250087098664502914694273, 8.866954213755877067428821723629, 9.939528895420697314900696333021