| L(s) = 1 | + 2.20i·5-s + i·7-s + 0.794·11-s − 5.12·13-s − 0.620i·17-s + 4i·19-s + 0.794·23-s + 0.123·25-s − 3.00i·29-s + 6.24i·31-s − 2.20·35-s − 11.1·37-s − 3.44i·41-s + 4i·43-s − 12.9·47-s + ⋯ |
| L(s) = 1 | + 0.987i·5-s + 0.377i·7-s + 0.239·11-s − 1.42·13-s − 0.150i·17-s + 0.917i·19-s + 0.165·23-s + 0.0246·25-s − 0.557i·29-s + 1.12i·31-s − 0.373·35-s − 1.82·37-s − 0.538i·41-s + 0.609i·43-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7074300953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7074300953\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.20iT - 5T^{2} \) |
| 11 | \( 1 - 0.794T + 11T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 + 0.620iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 0.794T + 23T^{2} \) |
| 29 | \( 1 + 3.00iT - 29T^{2} \) |
| 31 | \( 1 - 6.24iT - 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 3.00iT - 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 8.03T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 2.20iT - 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−9.677457754821136249448051998206, −8.723727730256298671062770731807, −7.899066344142551304725950409984, −7.04233881624308536254665827464, −6.57828458619212769010160284589, −5.50894160981427986772982658059, −4.76366116039857703831736263024, −3.54609219458057204734782369762, −2.79206764839168585375419470304, −1.77785082472166537681124319063,
0.24119135082338353605933630677, 1.55676736728367857165345996867, 2.75815839230473399436439565941, 3.93835656801716279213482337723, 4.88398491375367064824247721227, 5.24739205965284869655958935851, 6.55747455412167748089422661641, 7.20719748428391022992261910794, 8.059472441803635611578536614322, 8.837637163599885182949399347358
