L(s) = 1 | − i·2-s + 4-s + (1 − 2i)5-s − 2i·7-s − 3i·8-s + (−2 − i)10-s + 4·11-s − 2i·13-s − 2·14-s − 16-s + 4i·17-s − 19-s + (1 − 2i)20-s − 4i·22-s + 6i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (0.447 − 0.894i)5-s − 0.755i·7-s − 1.06i·8-s + (−0.632 − 0.316i)10-s + 1.20·11-s − 0.554i·13-s − 0.534·14-s − 0.250·16-s + 0.970i·17-s − 0.229·19-s + (0.223 − 0.447i)20-s − 0.852i·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08766 - 1.75988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08766 - 1.75988i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.895725449970585367293151930639, −9.378141900917385247922674040582, −8.300398733983534919188685853658, −7.32920404736049357396229373510, −6.39914183394441507505221523728, −5.53515279053185991094048963791, −4.16123666520399111066634718381, −3.51404870583363007594206669721, −1.90450570379301442420146239718, −1.07559288457100877122833086134,
1.96629034009164122585877834940, 2.79360950708595798475222006793, 4.20488842804589520322511362464, 5.63858215501018800429036408746, 6.13934856384532493371442001671, 7.01032123073199462453871565352, 7.53486084582928610991960209872, 8.956817079276231557468572180551, 9.272490544274385803527697612288, 10.58708562452459998415874618689