L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s − 4·11-s − i·12-s − 6i·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 − 1.66i·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
\(L(1)\) |
\(\approx\) |
\(1.189281772\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.189281772\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.342132617466661328431549257670, −8.081879826724826686626155483006, −7.30727041305469931317155355638, −6.17295461297151663563365913393, −5.66865917223226640709857301433, −5.01298742914042256728943913909, −4.08525513177826294622412090617, −3.28480431148627671517120141173, −2.24640206005644098778164929489, −0.47854328850114826098390521773,
0.823183893825295926897108568746, 2.19928422557990439415273858048, 2.55333156865305143807481482262, 3.80194338517480922719044684435, 4.67681722182461558600216132939, 5.35354256348786075248920017825, 6.39231239333070054480452787168, 7.11860118917362336569624195483, 7.77559413377405685528723924786, 8.747254709588423577340688029138