L(s) = 1 | + (−0.121 + 2.99i)3-s + 0.347i·5-s + 10.5·7-s + (−8.97 − 0.729i)9-s + 16.7i·11-s − 3.60·13-s + (−1.04 − 0.0423i)15-s + 27.6i·17-s − 22.1·19-s + (−1.28 + 31.6i)21-s − 32.7i·23-s + 24.8·25-s + (3.27 − 26.8i)27-s − 8.31i·29-s + 48.0·31-s + ⋯ |
L(s) = 1 | + (−0.0405 + 0.999i)3-s + 0.0695i·5-s + 1.51·7-s + (−0.996 − 0.0810i)9-s + 1.52i·11-s − 0.277·13-s + (−0.0695 − 0.00282i)15-s + 1.62i·17-s − 1.16·19-s + (−0.0612 + 1.50i)21-s − 1.42i·23-s + 0.995·25-s + (0.121 − 0.992i)27-s − 0.286i·29-s + 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0405 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0405 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.08840 + 1.04514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08840 + 1.04514i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.121 - 2.99i)T \) |
| 13 | \( 1 + 3.60T \) |
good | 5 | \( 1 - 0.347iT - 25T^{2} \) |
| 7 | \( 1 - 10.5T + 49T^{2} \) |
| 11 | \( 1 - 16.7iT - 121T^{2} \) |
| 17 | \( 1 - 27.6iT - 289T^{2} \) |
| 19 | \( 1 + 22.1T + 361T^{2} \) |
| 23 | \( 1 + 32.7iT - 529T^{2} \) |
| 29 | \( 1 + 8.31iT - 841T^{2} \) |
| 31 | \( 1 - 48.0T + 961T^{2} \) |
| 37 | \( 1 - 3.53T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 70.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 26.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 54.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 68.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 12.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 10.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 52.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 70.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.1T + 9.40e3T^{2} \) |
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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
−12.77402701048244758789341709859, −11.84683738042675368227939805945, −10.66993422253347610566829555821, −10.22424185868203591903022250422, −8.752762401296609667918493526220, −8.019809030073390134029535769052, −6.42108851861256615796534172791, −4.82615435424925963463863846310, −4.30570031532049057026805771579, −2.16820436151472294340094829721,
1.06546847504229657388565316791, 2.75765068411766563809168520989, 4.83561428504815834148974472841, 5.95519946749671064679328927234, 7.31622891239393450644180032989, 8.191811197486263664753142285732, 9.026673066842119911573625178806, 10.92025468945314552798650645383, 11.43056986436886538919383687330, 12.35794247197874735246560279332