L(s) = 1 | + (1.5 − 0.866i)5-s + (2.5 − 0.866i)7-s + (−4.5 − 2.59i)11-s − 6.92i·13-s + (−3 − 1.73i)17-s + (−1 − 1.73i)19-s + (−6 + 3.46i)23-s + (−1 + 1.73i)25-s + 9·29-s + (−0.5 + 0.866i)31-s + (3 − 3.46i)35-s + (1 + 1.73i)37-s − 3.46i·41-s + 3.46i·43-s + (5.5 − 4.33i)49-s + ⋯ |
L(s) = 1 | + (0.670 − 0.387i)5-s + (0.944 − 0.327i)7-s + (−1.35 − 0.783i)11-s − 1.92i·13-s + (−0.727 − 0.420i)17-s + (−0.229 − 0.397i)19-s + (−1.25 + 0.722i)23-s + (−0.200 + 0.346i)25-s + 1.67·29-s + (−0.0898 + 0.155i)31-s + (0.507 − 0.585i)35-s + (0.164 + 0.284i)37-s − 0.541i·41-s + 0.528i·43-s + (0.785 − 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.564373530\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564373530\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6 - 3.46i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.92iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−9.975045519953463060562262920260, −8.730587142109839978131006742470, −8.100030970195245716863865335897, −7.50530572768532093589279947603, −6.08832961564354635157744949681, −5.36117816755152438890553138862, −4.74839696643501832681308898618, −3.25317360801401223048450108988, −2.20468572041481814113360409137, −0.68409286285042670480214725991,
1.98003035227879247281892375450, 2.35382533684508749216753455626, 4.25638641070291280378823536336, 4.80945232585049918090712002554, 6.02551987270255980345549577264, 6.68284493660494082509932457183, 7.76943450381297701986946427752, 8.482494123929145211904405253515, 9.392456913647069536027902577335, 10.30137223448153356439117196703