L(s) = 1 | + 2.82i·5-s + 4·7-s − 1.41i·11-s − 2.82i·13-s + 4·17-s + 7.07i·19-s − 4·23-s − 3.00·25-s + 8.48i·29-s − 8·31-s + 11.3i·35-s + 2.82i·37-s + 2·41-s + 4.24i·43-s + 9·49-s + ⋯ |
L(s) = 1 | + 1.26i·5-s + 1.51·7-s − 0.426i·11-s − 0.784i·13-s + 0.970·17-s + 1.62i·19-s − 0.834·23-s − 0.600·25-s + 1.57i·29-s − 1.43·31-s + 1.91i·35-s + 0.464i·37-s + 0.312·41-s + 0.646i·43-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.207807463\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.207807463\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.07iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 8.48iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.161048167499630293320706591908, −7.85707408947901313955270006254, −7.24084011673961309058733793923, −6.23759932069706447639389250404, −5.59278205061334039699439425990, −4.94967096528404641915530265275, −3.71205497990449823282899726602, −3.27918955886541564057680195097, −2.12474932894076248211621500644, −1.27947935886242528408816348254,
0.63473727976544485320925178187, 1.65741733436685305257799002990, 2.33115270495740586752525859178, 3.90834311019608870711795833723, 4.48786736240964925000394711254, 5.11254444484965708242530121742, 5.65824006542827528276795866023, 6.78719626837114206837172959974, 7.69053922091741220420945393274, 8.033404785342362874929681190587