| L(s) = 1 | − 0.646·5-s − i·7-s − 3.09i·11-s − 0.913i·13-s + 6.77i·17-s − 5.29·19-s − 2.53·23-s − 4.58·25-s + 9.93·29-s + 9.16i·31-s + 0.646i·35-s + 7.84i·37-s − 9.01i·41-s + 12.2·43-s + 5.65·47-s + ⋯ |
| L(s) = 1 | − 0.288·5-s − 0.377i·7-s − 0.933i·11-s − 0.253i·13-s + 1.64i·17-s − 1.21·19-s − 0.528·23-s − 0.916·25-s + 1.84·29-s + 1.64i·31-s + 0.109i·35-s + 1.28i·37-s − 1.40i·41-s + 1.86·43-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.507127186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507127186\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.646T + 5T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 + 0.913iT - 13T^{2} \) |
| 17 | \( 1 - 6.77iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 - 9.93T + 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 + 9.01iT - 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 1.15T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 0.417T + 73T^{2} \) |
| 79 | \( 1 + 10.7iT - 79T^{2} \) |
| 83 | \( 1 + 15.9iT - 83T^{2} \) |
| 89 | \( 1 - 9.01iT - 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−8.439606709565785461054617027748, −7.953083026104907420236843749133, −6.96339741013036181580944582763, −6.22419387014833846617187556300, −5.70023631736672270368403524011, −4.53127513792266685024996674021, −3.92857325406980701901552374691, −3.12421944071401971005400486475, −1.99713703496434975495584851528, −0.816363954498810292737362315756,
0.58883657103295071608592768301, 2.16157982620422420272345500687, 2.64445839925765612367423455824, 4.05724690543260640475701009415, 4.45880429627601369538802227373, 5.41297959514633919737207275293, 6.25182774301847625845359576642, 6.95570712915239493604691294538, 7.72814263323340807146586914336, 8.269718339952688233470327608716
