| L(s) = 1 | − i·3-s + i·7-s − 9-s − 6·11-s − 2i·13-s + 4·19-s + 21-s + 6i·23-s + i·27-s − 6·29-s + 8·31-s + 6i·33-s + 2i·37-s − 2·39-s + 12·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.80·11-s − 0.554i·13-s + 0.917·19-s + 0.218·21-s + 1.25i·23-s + 0.192i·27-s − 1.11·29-s + 1.43·31-s + 1.04i·33-s + 0.328i·37-s − 0.320·39-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068786012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068786012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.334160588088703665723149310247, −8.096362834242939938755733891477, −7.84588758143579943271053413032, −7.09959069637711725499202161259, −5.79532631877383755956047939166, −5.56894161311508485763590873490, −4.50078541102630895590818772348, −3.07240320708764163054184539503, −2.56113527066698518566206620410, −1.16120108529434125444805543616,
0.40506453482796761740976323754, 2.20418094039544649709065316617, 3.07919939410614854098454602045, 4.13766361371917066456946450974, 4.94175526113973912105672302566, 5.61235911729251425840469039932, 6.62069975807823456338094688079, 7.58239115349956445927609861427, 8.112979180081501232426812543219, 9.056382081793747333227466982444
