| L(s) = 1 | + 0.528·2-s − 3.72·4-s + (−3.86 − 3.17i)5-s − 2.76i·7-s − 4.08·8-s + (−2.04 − 1.67i)10-s + 9.22i·11-s + 13.5i·13-s − 1.46i·14-s + 12.7·16-s + 12.2·17-s + 20.2·19-s + (14.3 + 11.8i)20-s + 4.87i·22-s − 2.37·23-s + ⋯ |
| L(s) = 1 | + 0.264·2-s − 0.930·4-s + (−0.772 − 0.635i)5-s − 0.395i·7-s − 0.510·8-s + (−0.204 − 0.167i)10-s + 0.838i·11-s + 1.04i·13-s − 0.104i·14-s + 0.795·16-s + 0.718·17-s + 1.06·19-s + (0.718 + 0.590i)20-s + 0.221i·22-s − 0.103·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.977326 + 0.461583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.977326 + 0.461583i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (3.86 + 3.17i)T \) |
| good | 2 | \( 1 - 0.528T + 4T^{2} \) |
| 7 | \( 1 + 2.76iT - 49T^{2} \) |
| 11 | \( 1 - 9.22iT - 121T^{2} \) |
| 13 | \( 1 - 13.5iT - 169T^{2} \) |
| 17 | \( 1 - 12.2T + 289T^{2} \) |
| 19 | \( 1 - 20.2T + 361T^{2} \) |
| 23 | \( 1 + 2.37T + 529T^{2} \) |
| 29 | \( 1 + 34.9iT - 841T^{2} \) |
| 31 | \( 1 - 29.4T + 961T^{2} \) |
| 37 | \( 1 - 64.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 93.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 9.82T + 2.80e3T^{2} \) |
| 59 | \( 1 + 58.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 15.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 53.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 34.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 29.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 62.3iT - 9.40e3T^{2} \) |
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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
−11.55839092289103585112036409616, −9.873538219482785125420704347692, −9.547703389023313841391258264639, −8.296913468888677363817711228631, −7.68583087180890248697183594916, −6.39387312629353998411251640846, −4.91825296719678980211964266067, −4.45299065436696104772965110567, −3.33655050980172839946557944167, −1.14190756585098509647517924035,
0.55290005114817471733578698517, 3.05597149594692226621097794065, 3.70050949951061813246907702289, 5.14373881942763619319840519536, 5.87697793541532263123318881586, 7.32081873247391858218649993275, 8.181641386799609595158124945795, 8.961428629966999745135131031020, 10.09614879870938312443362291620, 10.88321160822630065094260789056
