| L(s) = 1 | + (−2 + i)5-s + 4i·13-s − 2i·17-s + (3 − 4i)25-s − 4·29-s + 12i·37-s − 8·41-s + 7·49-s − 14i·53-s − 10·61-s + (−4 − 8i)65-s − 16i·73-s + (2 + 4i)85-s − 16·89-s − 8i·97-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.447i)5-s + 1.10i·13-s − 0.485i·17-s + (0.600 − 0.800i)25-s − 0.742·29-s + 1.97i·37-s − 1.24·41-s + 49-s − 1.92i·53-s − 1.28·61-s + (−0.496 − 0.992i)65-s − 1.87i·73-s + (0.216 + 0.433i)85-s − 1.69·89-s − 0.812i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 12iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.398068155268665612867394609022, −7.68906851877904763743470860461, −6.88589820537729023502123313867, −6.44941257269850823609210940526, −5.22419444059934762524224117609, −4.45576074127851331169903469618, −3.65499272545099430131064274293, −2.81217713490739713776022630972, −1.61197091220539638492840631325, 0,
1.25244491568030934726458751790, 2.64025610554223061129127365949, 3.64228590446955423322009395276, 4.26631858909385865003102086311, 5.31575906894959062907012800200, 5.85758768358928873174161019012, 7.06160305429228138953223555934, 7.61897113993933703535736158824, 8.323587482551749396540417152965
