L(s) = 1 | + (0.5 + 2.95i)3-s + 8·7-s + (−8.5 + 2.95i)9-s − 17.7i·11-s − 2·13-s + 17.7i·17-s − 11·19-s + (4 + 23.6i)21-s + 35.4i·23-s + (−13 − 23.6i)27-s + 35.4i·29-s + 46·31-s + (52.5 − 8.87i)33-s + 16·37-s + (−1 − 5.91i)39-s + ⋯ |
L(s) = 1 | + (0.166 + 0.986i)3-s + 1.14·7-s + (−0.944 + 0.328i)9-s − 1.61i·11-s − 0.153·13-s + 1.04i·17-s − 0.578·19-s + (0.190 + 1.12i)21-s + 1.54i·23-s + (−0.481 − 0.876i)27-s + 1.22i·29-s + 1.48·31-s + (1.59 − 0.268i)33-s + 0.432·37-s + (−0.0256 − 0.151i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.044046289\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.044046289\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 2.95i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 8T + 49T^{2} \) |
| 11 | \( 1 + 17.7iT - 121T^{2} \) |
| 13 | \( 1 + 2T + 169T^{2} \) |
| 17 | \( 1 - 17.7iT - 289T^{2} \) |
| 19 | \( 1 + 11T + 361T^{2} \) |
| 23 | \( 1 - 35.4iT - 529T^{2} \) |
| 29 | \( 1 - 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 46T + 961T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 62T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 70.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113T + 4.48e3T^{2} \) |
| 71 | \( 1 - 106. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 101T + 5.32e3T^{2} \) |
| 79 | \( 1 + 68T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22T + 9.40e3T^{2} \) |
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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
−9.850593898804046294539661728391, −8.729308228141334049619387225024, −8.445714136487085271556949721215, −7.59903130782252000962467945800, −6.13352289307853883002477016007, −5.51654733592879906091759237707, −4.57271142789805412096121521407, −3.72404214635462295647506257984, −2.74543782384356575426931918215, −1.25478643712709726228180034542,
0.63629568886352041505168016092, 2.00170341477063596083510680171, 2.56127366772225805873811755341, 4.34345649447911130584429053018, 4.91749018845847083820293719435, 6.16244070696292573157067135268, 6.98276980673596953145490801971, 7.69802484254808182178168583935, 8.291703415065667236211511152864, 9.220392785961744204865643940803