L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.499 − 0.866i)6-s + (1.78 + 3.08i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.280 − 0.486i)11-s + 0.999·12-s + (2.84 + 2.21i)13-s − 3.56·14-s + (−0.5 + 0.866i)16-s + (1.56 + 2.70i)17-s + 0.999·18-s + (1.21 + 2.11i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.204 − 0.353i)6-s + (0.673 + 1.16i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.0846 − 0.146i)11-s + 0.288·12-s + (0.788 + 0.615i)13-s − 0.951·14-s + (−0.125 + 0.216i)16-s + (0.378 + 0.655i)17-s + 0.235·18-s + (0.279 + 0.484i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.447478489\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.447478489\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.84 - 2.21i)T \) |
good | 7 | \( 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.280 + 0.486i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 2.70i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.21 - 2.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.84 + 6.65i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.561 - 0.972i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (0.280 - 0.486i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.56 + 2.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.842 - 1.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.90 - 10.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.28 + 10.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 6.56T + 83T^{2} \) |
| 89 | \( 1 + (5.12 - 8.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.40 - 2.43i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.151615281861327952083884363754, −8.702512479662282610747545823674, −8.162182174250796946559074754440, −7.04305008290001033120398817879, −6.13366966288620055430666944235, −5.62624651169231994252144084049, −4.76214168797417561251119486865, −3.88831806205572272985996982160, −2.54285196020158346336533423871, −1.24328915854516000488841986346,
0.75723357489625124904657591091, 1.47271067482213714227876390537, 2.87049496051891924484942461643, 3.80294833043411453821392761823, 4.78765305051872420537478003421, 5.60970730723455523042275500467, 6.80862558046174272395183774030, 7.49060626564070976323470121094, 8.006144618475602204073708783732, 8.938108550006578133921567583854