L(s) = 1 | − 1.51·5-s − i·7-s + 0.935i·11-s + 2.14i·13-s − 2.62i·17-s − 3.46·19-s + 6.86·23-s − 2.70·25-s + 2.44·29-s + 2i·31-s + 1.51i·35-s − 2.14i·37-s − 2.62i·41-s − 7.74·43-s − 49-s + ⋯ |
L(s) = 1 | − 0.677·5-s − 0.377i·7-s + 0.282i·11-s + 0.593i·13-s − 0.635i·17-s − 0.794·19-s + 1.43·23-s − 0.541·25-s + 0.454·29-s + 0.359i·31-s + 0.255i·35-s − 0.351i·37-s − 0.409i·41-s − 1.18·43-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.280147492\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.280147492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 11 | \( 1 - 0.935iT - 11T^{2} \) |
| 13 | \( 1 - 2.14iT - 13T^{2} \) |
| 17 | \( 1 + 2.62iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 6.86T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 2.14iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 7.74T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.79iT - 59T^{2} \) |
| 61 | \( 1 - 11.2iT - 61T^{2} \) |
| 67 | \( 1 - 2.14T + 67T^{2} \) |
| 71 | \( 1 - 1.62T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 + 1.70iT - 79T^{2} \) |
| 83 | \( 1 - 1.87iT - 83T^{2} \) |
| 89 | \( 1 - 2.62iT - 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.672937249366213199910703870000, −7.67588548450207544698763448499, −7.12739417923489697711242389828, −6.55744434362851169239679315491, −5.48998489883821765778825778274, −4.64397601303872561793044052611, −4.05266995340931193957542161739, −3.15660190781503459765933654523, −2.13846720319136633956669556120, −0.875629124855549268459019402381,
0.47750488155495991797549920577, 1.83408002363489335005631786424, 2.95465392820715264311468793404, 3.65068627548962566500434773878, 4.56907448356319308325199968782, 5.32193350359249451300449433112, 6.18289523959380149802927805671, 6.84830078816260027912200688538, 7.75488373513653403107904966849, 8.344006896639907171206882570982