L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (2.73 + 2.73i)7-s + (0.707 + 0.707i)8-s − 1.99i·9-s + (−0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s − 3.87·14-s − 1.00·16-s + (2.73 + 2.73i)17-s + (1.41 + 1.41i)18-s + (−3.87 + 2i)19-s − 3.87i·21-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + 0.408·6-s + (1.03 + 1.03i)7-s + (0.250 + 0.250i)8-s − 0.666i·9-s + (−0.204 + 0.204i)12-s + (0.196 + 0.196i)13-s − 1.03·14-s − 0.250·16-s + (0.664 + 0.664i)17-s + (0.333 + 0.333i)18-s + (−0.888 + 0.458i)19-s − 0.845i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10672 + 0.409328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10672 + 0.409328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.87 - 2i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.73 - 2.73i)T + 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.73 - 2.73i)T + 17iT^{2} \) |
| 23 | \( 1 + (-2.73 + 2.73i)T - 23iT^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-5.47 - 5.47i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.36 - 6.36i)T + 53iT^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + (9.19 - 9.19i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.74iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 2.73i)T - 73iT^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + (-5.47 + 5.47i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + (-5.65 + 5.65i)T - 97iT^{2} \) |
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
−10.14799933985077336652682159208, −8.902130174631104185676313649270, −8.611769818399855152658659211264, −7.66119973828738473059604862694, −6.68020067482736616899931726498, −5.92433401289976454217290021867, −5.23623282179349420854830818158, −4.00386566053095127087883551686, −2.32944653671989894751878500588, −1.13608758998098718595454161644,
0.872906020208724217421846810586, 2.22103519689679304277763265989, 3.64909703253174139461402082186, 4.64616092224960490087168494339, 5.27996846013383771615769632748, 6.73948936944275147149023126516, 7.65664114990535511496113153918, 8.213707831476766140667852723731, 9.239242899384424303856610673682, 10.30042489685193486754896540351