| L(s) = 1 | + 5-s − 4.17i·7-s + 4.20·11-s + 6.43·13-s + 6.40·17-s + 3.39i·19-s + (4.53 − 1.56i)23-s + 25-s − 4.78i·29-s − 4.55·31-s − 4.17i·35-s + 0.928i·37-s − 5.78i·41-s + 4.60i·43-s + 9.84i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.57i·7-s + 1.26·11-s + 1.78·13-s + 1.55·17-s + 0.777i·19-s + (0.945 − 0.325i)23-s + 0.200·25-s − 0.888i·29-s − 0.817·31-s − 0.705i·35-s + 0.152i·37-s − 0.902i·41-s + 0.702i·43-s + 1.43i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.816993044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.816993044\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + (-4.53 + 1.56i)T \) |
| good | 7 | \( 1 + 4.17iT - 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 6.43T + 13T^{2} \) |
| 17 | \( 1 - 6.40T + 17T^{2} \) |
| 19 | \( 1 - 3.39iT - 19T^{2} \) |
| 29 | \( 1 + 4.78iT - 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 - 0.928iT - 37T^{2} \) |
| 41 | \( 1 + 5.78iT - 41T^{2} \) |
| 43 | \( 1 - 4.60iT - 43T^{2} \) |
| 47 | \( 1 - 9.84iT - 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 10.7iT - 61T^{2} \) |
| 67 | \( 1 - 3.60iT - 67T^{2} \) |
| 71 | \( 1 - 1.57iT - 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 5.66T + 89T^{2} \) |
| 97 | \( 1 + 17.8iT - 97T^{2} \) |
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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
−8.301989185741819832801300057459, −7.57996913827933934033045672312, −6.86694286918386677673228451607, −6.15602745031594876145996541614, −5.57155873193138873734714083604, −4.25898270016418376711781646714, −3.86197067080144006367361187135, −3.09396230639585512091058775924, −1.30363692325153320515299961402, −1.15846244200392078813733155165,
1.16799646932891193979214366848, 1.92451280895020239920752377629, 3.20542111816196843280558446687, 3.60848757465024734320170423412, 5.04039220415416301505828359648, 5.51316886570346615888697506602, 6.30299888139528331646655027650, 6.75637287212402047745066710744, 7.937800034680488250041698770335, 8.684455238655107088074165493625
