| L(s) = 1 | + (0.353 − 2.20i)5-s − 3.09i·7-s − 5.51·11-s − i·13-s + 6.21i·17-s + 4.21i·23-s + (−4.74 − 1.56i)25-s − 1.70·29-s + (−6.83 − 1.09i)35-s − 4.02i·37-s − 0.198·41-s + 5.70i·43-s + 6.41i·47-s − 2.57·49-s + 4.02i·53-s + ⋯ |
| L(s) = 1 | + (0.158 − 0.987i)5-s − 1.16i·7-s − 1.66·11-s − 0.277i·13-s + 1.50i·17-s + 0.879i·23-s + (−0.949 − 0.312i)25-s − 0.317·29-s + (−1.15 − 0.184i)35-s − 0.662i·37-s − 0.0310·41-s + 0.870i·43-s + 0.935i·47-s − 0.367·49-s + 0.553i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5584094259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5584094259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.353 + 2.20i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.09iT - 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 17 | \( 1 - 6.21iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.21iT - 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.02iT - 37T^{2} \) |
| 41 | \( 1 + 0.198T + 41T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 - 6.41iT - 47T^{2} \) |
| 53 | \( 1 - 4.02iT - 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 + 9.31iT - 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 + 9.40T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
| 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
−8.283621694261311484218557970160, −7.79703632304631467096589533123, −7.29154840730189495012894235530, −6.07459551211222155329967739342, −5.56660840279795079281889014093, −4.71540654203190651478252818555, −4.08086581760204120637850612915, −3.18845522336633343182696634411, −1.98849853011643735788114876531, −1.03114653801596444840217503826,
0.16171433733128917064349804550, 2.10745206450652503191027470819, 2.63002211316516618451668245692, 3.23902353012138042802153245786, 4.54981630787071468467903192894, 5.38270220960085891133909624371, 5.78871560778109443905930282362, 6.85369981299194746459897648838, 7.26477856522242399134615010445, 8.210897905378085306691948415950
