L(s) = 1 | − 1.73i·2-s − 1.73i·3-s − 0.999·4-s − 5-s − 2.99·6-s + (2 + 1.73i)7-s − 1.73i·8-s − 2.99·9-s + 1.73i·10-s + 3.46i·11-s + 1.73i·12-s + (2.99 − 3.46i)14-s + 1.73i·15-s − 5·16-s + 6·17-s + 5.19i·18-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.999i·3-s − 0.499·4-s − 0.447·5-s − 1.22·6-s + (0.755 + 0.654i)7-s − 0.612i·8-s − 0.999·9-s + 0.547i·10-s + 1.04i·11-s + 0.499i·12-s + (0.801 − 0.925i)14-s + 0.447i·15-s − 1.25·16-s + 1.45·17-s + 1.22i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.432379 - 0.946436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.432379 - 0.946436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−12.82009214947011327893698412020, −12.26108891185029541558057106982, −11.58250802746344809847045416066, −10.55467498088450060964845520155, −9.150575487050004757995450630884, −7.899835355812656162524348600013, −6.77936700043949431351337836028, −4.96741107106739787117608112172, −3.02106091710520419118583338891, −1.60623861147945947594658943182,
3.63401856002163203666662374263, 5.05964223157965452541194320082, 6.07563044325703369545735779630, 7.78988597368559588077758915153, 8.253625310433564150149755056434, 9.769668520548840763317123178411, 11.02441236922778163076569590742, 11.73201949777095029447336782148, 13.75691866366674629273059850121, 14.44085454038547082654550860093