| L(s) = 1 | + (1 − i)3-s + (−1 + 2i)5-s + i·9-s + (−3 + 3i)11-s + (−3 + 3i)13-s + (1 + 3i)15-s + 4i·17-s + (−1 − i)19-s + 8·23-s + (−3 − 4i)25-s + (4 + 4i)27-s + (−3 − 3i)29-s + 6i·33-s + (−3 − 3i)37-s + 6i·39-s + ⋯ |
| L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.447 + 0.894i)5-s + 0.333i·9-s + (−0.904 + 0.904i)11-s + (−0.832 + 0.832i)13-s + (0.258 + 0.774i)15-s + 0.970i·17-s + (−0.229 − 0.229i)19-s + 1.66·23-s + (−0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s + (−0.557 − 0.557i)29-s + 1.04i·33-s + (−0.493 − 0.493i)37-s + 0.960i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.920166 + 0.857093i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.920166 + 0.857093i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + (1 + i)T + 19iT^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9 + 9i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5 - 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (3 - 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-9 + 9i)T - 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−10.74882932097001688569872328489, −10.02422947359015508493052249545, −8.931900027358690525427905030296, −7.934188730423437248325410580756, −7.28853589815066075941469339499, −6.72687674706282178522360407733, −5.24512710445198219723813836794, −4.16645413286408649562553726047, −2.78832385338468890164191577751, −2.03699737882536283037917391263,
0.62029260297903430405597842059, 2.78011231566836806873499945014, 3.60532436475525668714739903053, 4.90268197831600952299765836974, 5.43700459579989669504937424426, 6.99797587148800500169818590833, 7.967965778865657239327681287075, 8.696290961091251964030592508103, 9.364282824778839666626575241565, 10.24428104584806873137314125239
