| L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 3.44i·7-s − i·8-s + 9-s − 10-s + 4.24i·11-s − 12-s − 3.44·14-s + i·15-s + 16-s − 6.78·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 1.30i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.28i·11-s − 0.288·12-s − 0.920·14-s + 0.258i·15-s + 0.250·16-s − 1.64·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5624765878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5624765878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3.44iT - 7T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 + 6.26iT - 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 + 9.14T + 29T^{2} \) |
| 31 | \( 1 + 3.75iT - 31T^{2} \) |
| 37 | \( 1 - 6.82iT - 37T^{2} \) |
| 41 | \( 1 + 4.26iT - 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 - 7.76iT - 47T^{2} \) |
| 53 | \( 1 + 8.93T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 67 | \( 1 - 0.0760iT - 67T^{2} \) |
| 71 | \( 1 + 0.374iT - 71T^{2} \) |
| 73 | \( 1 + 16.7iT - 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 0.740iT - 83T^{2} \) |
| 89 | \( 1 - 13.3iT - 89T^{2} \) |
| 97 | \( 1 - 13.1iT - 97T^{2} \) |
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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
−8.777039633545669151737727377366, −7.971802823074793637043990248409, −7.25266339075493303420993551798, −6.67816207451123191627810901604, −6.04243152371806452487416290634, −4.94752695081701915989454958706, −4.59701464751868452104550860869, −3.46659737160920582675852300934, −2.44882212093264362702222610065, −1.96173845377314467885795007339,
0.13368404916328274429599631875, 1.24307363973396128094039139024, 2.09394379201956295066445672341, 3.29381609588877298272604138672, 3.83363572945920987749622210381, 4.41651198201354329779915296581, 5.41606495348098834504643395700, 6.25584553425688102067882840385, 7.21263601307506748894325141957, 7.81998404904056744857088585831
