| L(s) = 1 | + 0.810·3-s − 3.82i·5-s + i·7-s − 2.34·9-s − i·11-s + (3.12 + 1.79i)13-s − 3.09i·15-s + 1.91·17-s + 2.75i·19-s + 0.810i·21-s − 7.88·23-s − 9.63·25-s − 4.32·27-s − 8.14·29-s − 6.00i·31-s + ⋯ |
| L(s) = 1 | + 0.467·3-s − 1.71i·5-s + 0.377i·7-s − 0.781·9-s − 0.301i·11-s + (0.866 + 0.498i)13-s − 0.800i·15-s + 0.463·17-s + 0.631i·19-s + 0.176i·21-s − 1.64·23-s − 1.92·25-s − 0.833·27-s − 1.51·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2861801964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2861801964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-3.12 - 1.79i)T \) |
| good | 3 | \( 1 - 0.810T + 3T^{2} \) |
| 5 | \( 1 + 3.82iT - 5T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 - 2.75iT - 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 + 6.00iT - 31T^{2} \) |
| 37 | \( 1 + 5.83iT - 37T^{2} \) |
| 41 | \( 1 - 5.43iT - 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 7.25T + 53T^{2} \) |
| 59 | \( 1 + 10.5iT - 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 1.53iT - 67T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 9.21iT - 73T^{2} \) |
| 79 | \( 1 + 6.37T + 79T^{2} \) |
| 83 | \( 1 + 4.00iT - 83T^{2} \) |
| 89 | \( 1 - 8.61iT - 89T^{2} \) |
| 97 | \( 1 + 1.03iT - 97T^{2} \) |
| 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
−8.040090481488510289062439284922, −7.83541998214833088605500441966, −6.17463314453843432547676645211, −5.83637050062802321227994986041, −5.09065321412139579473117944583, −4.07146357829540323590038775638, −3.55690200465890695298545746603, −2.19533201955950676396598804753, −1.42499038297170053352295758856, −0.07099745035160964689717049169,
1.80314162834353908653081489256, 2.70508702080332526116766530984, 3.44635458673188638011024678644, 3.90049342609648054508202430183, 5.35451000128519326402716265118, 6.01289306584414816768992063805, 6.78373840485538389613542364942, 7.36915579538683102498874378775, 8.104650014474620529720889212947, 8.711509276805356094410651873418
