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 | \( 1 - 2.73iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.157538561518016868100840073072, −8.560917974555283032861511261437, −7.65054818794943032688226388979, −6.64648546262920849764761053320, −5.76772495241756239337781452461, −4.78947517201624457546203717413, −4.20538140682006614814992807258, −3.26144314545617586905436109413, −2.72345909511125749642439600407, −0.20083045312349155890648141678,
1.42130856727910920820615895387, 1.75507164389864830522404511013, 3.35293603993835951572298733275, 4.28603660174714292098384000101, 5.47391069713089644258351766706, 6.31875943218937970602722187399, 6.93232773167095691843879647872, 7.66195133692207229806815232171, 8.468115596384972432781162046237, 8.804428002637994932192605069137