| L(s) = 1 | − 3.12i·3-s + (−1.32 − 1.80i)5-s − 6.76·9-s + 2.48·11-s + 4.15i·13-s + (−5.64 + 4.12i)15-s + 5.76i·17-s − 1.60·19-s + 7.28i·23-s + (−1.51 + 4.76i)25-s + 11.7i·27-s − 1.45·29-s + 2.24·31-s − 7.76i·33-s + 6i·37-s + ⋯ |
| L(s) = 1 | − 1.80i·3-s + (−0.590 − 0.807i)5-s − 2.25·9-s + 0.749·11-s + 1.15i·13-s + (−1.45 + 1.06i)15-s + 1.39i·17-s − 0.369·19-s + 1.51i·23-s + (−0.303 + 0.952i)25-s + 2.26i·27-s − 0.270·29-s + 0.404·31-s − 1.35i·33-s + 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5511882624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5511882624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.32 + 1.80i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.12iT - 3T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.15iT - 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 7.28iT - 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.28iT - 43T^{2} \) |
| 47 | \( 1 - 3.45iT - 47T^{2} \) |
| 53 | \( 1 + 9.21iT - 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 + 5.35T + 61T^{2} \) |
| 67 | \( 1 - 7.52iT - 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 7.28iT - 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 10.1iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.73iT - 97T^{2} \) |
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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
−8.804428002637994932192605069137, −8.468115596384972432781162046237, −7.66195133692207229806815232171, −6.93232773167095691843879647872, −6.31875943218937970602722187399, −5.47391069713089644258351766706, −4.28603660174714292098384000101, −3.35293603993835951572298733275, −1.75507164389864830522404511013, −1.42130856727910920820615895387,
0.20083045312349155890648141678, 2.72345909511125749642439600407, 3.26144314545617586905436109413, 4.20538140682006614814992807258, 4.78947517201624457546203717413, 5.76772495241756239337781452461, 6.64648546262920849764761053320, 7.65054818794943032688226388979, 8.560917974555283032861511261437, 9.157538561518016868100840073072
