L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.55 − 0.759i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−1.63 + 0.564i)6-s + 1.92·7-s + (−0.707 − 0.707i)8-s + (1.84 + 2.36i)9-s + 1.00·10-s − 5.18·11-s + (−0.759 + 1.55i)12-s + (−3.96 + 3.96i)13-s + (1.36 − 1.36i)14-s + (−0.564 − 1.63i)15-s − 1.00·16-s + (−3.83 − 3.83i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.898 − 0.438i)3-s − 0.500i·4-s + (0.316 + 0.316i)5-s + (−0.668 + 0.230i)6-s + 0.727·7-s + (−0.250 − 0.250i)8-s + (0.615 + 0.787i)9-s + 0.316·10-s − 1.56·11-s + (−0.219 + 0.449i)12-s + (−1.10 + 1.10i)13-s + (0.363 − 0.363i)14-s + (−0.145 − 0.422i)15-s − 0.250·16-s + (−0.930 − 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02323936777\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02323936777\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.55 + 0.759i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 37 | \( 1 + (5.86 - 1.59i)T \) |
good | 7 | \( 1 - 1.92T + 7T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 13 | \( 1 + (3.96 - 3.96i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.83 + 3.83i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0989 + 0.0989i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.07 + 1.07i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.77 - 1.77i)T - 29iT^{2} \) |
| 31 | \( 1 + (4.45 + 4.45i)T + 31iT^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + (-7.21 + 7.21i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.47iT - 47T^{2} \) |
| 53 | \( 1 + 3.95iT - 53T^{2} \) |
| 59 | \( 1 + (-7.83 - 7.83i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.10 - 9.10i)T + 61iT^{2} \) |
| 67 | \( 1 - 9.29iT - 67T^{2} \) |
| 71 | \( 1 - 1.23iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + (0.761 - 0.761i)T - 79iT^{2} \) |
| 83 | \( 1 - 9.18iT - 83T^{2} \) |
| 89 | \( 1 + (5.54 - 5.54i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.84 + 3.84i)T - 97iT^{2} \) |
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
−9.659008488321917646266191327271, −8.464087720957379589886910582697, −7.24758157691364446127166105897, −6.92776609826215772710007393653, −5.55301685582091568467434928041, −5.10840012219184625046619442154, −4.28410890132736796092439217143, −2.53434556041901627667870308788, −1.91237410272164710517859141826, −0.008751530301217760471444672518,
2.08018936118948402378859109406, 3.50716082057844731647702442537, 4.83711587971632407093240079383, 5.11946302312483160859015690279, 5.85422999969237283205874026711, 6.94138051494441678976782309322, 7.82817388889053632270590493425, 8.522660839820770857783391762922, 9.737650521146752220612796285947, 10.47771538106142668169935262905