L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + i·8-s − 9-s + 4·11-s − i·12-s − 2i·13-s + 16-s + 6i·17-s + i·18-s − 4·19-s − 4i·22-s − i·23-s − 24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 1.45i·17-s + 0.235i·18-s − 0.917·19-s − 0.852i·22-s − 0.208i·23-s − 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.656484061\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.656484061\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.721257612672303928240451252967, −8.277454573085354693538528366381, −7.18346000723929198360800345560, −6.16997369161889588760328174383, −5.67162440939046650070474846412, −4.37743677480524389349108546609, −4.10507696503368596181639656498, −3.16621718586258941610371833445, −2.14233836274020771822963299064, −1.02454355335564906969862076797,
0.60592267372715851636262653171, 1.81064149604612219702921404378, 2.95513192686867993233073079223, 4.08255329946938097738527943617, 4.72233001461290102610888466594, 5.76683707148325458126071552406, 6.43325810025645327369490292698, 7.00900165194801252251388128122, 7.62533123138108109294223174579, 8.487612727507937873567076968614