L(s) = 1 | + (−1.58 + 1.14i)5-s + (−0.413 − 1.27i)7-s + (0.809 + 0.587i)9-s + (−0.669 − 0.743i)11-s + (0.478 + 0.658i)17-s + (0.309 − 0.951i)19-s + 1.17i·23-s + (0.873 − 2.68i)25-s + (2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (1.89 + 0.614i)47-s + (−0.639 + 0.464i)49-s + (1.91 + 0.406i)55-s + (1.16 + 1.60i)61-s + ⋯ |
L(s) = 1 | + (−1.58 + 1.14i)5-s + (−0.413 − 1.27i)7-s + (0.809 + 0.587i)9-s + (−0.669 − 0.743i)11-s + (0.478 + 0.658i)17-s + (0.309 − 0.951i)19-s + 1.17i·23-s + (0.873 − 2.68i)25-s + (2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (1.89 + 0.614i)47-s + (−0.639 + 0.464i)49-s + (1.91 + 0.406i)55-s + (1.16 + 1.60i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8669924687\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8669924687\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (1.58 - 1.14i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.413 + 1.27i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.478 - 0.658i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 1.17iT - T^{2} \) |
| 29 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 0.209T + T^{2} \) |
| 47 | \( 1 + (-1.89 - 0.614i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.16 - 1.60i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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
−8.584333781025771616741604381427, −7.75809793283748588288445724211, −7.37836541940384448620671872623, −6.98709178454556908116914187573, −5.96092182642097336666927472231, −4.78067915557903445617919172644, −3.92241392754842020571529557273, −3.51940269477971777897601783698, −2.59859775954032329717097412120, −0.830770574104273442470869199996,
0.804402916701645005967429567367, 2.24519552047240034673382469352, 3.40951981496592452168893676388, 4.12875991739678432403139347053, 4.94220372051221529316674764336, 5.52184290270018919739624088233, 6.66079065248697834635048018878, 7.47063829236996854494451382206, 8.042309278852142393184589272110, 8.740264213115159813851260013079