| L(s) = 1 | − i·3-s − i·5-s − 2.40·7-s − 9-s + 2.34·11-s − 5.36·13-s − 15-s + 0.425i·17-s − 3.47·19-s + 2.40i·21-s + (0.874 − 4.71i)23-s − 25-s + i·27-s + 4.06·29-s + 10.3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s − 0.908·7-s − 0.333·9-s + 0.706·11-s − 1.48·13-s − 0.258·15-s + 0.103i·17-s − 0.796·19-s + 0.524i·21-s + (0.182 − 0.983i)23-s − 0.200·25-s + 0.192i·27-s + 0.754·29-s + 1.86i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.020051151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.020051151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-0.874 + 4.71i)T \) |
| good | 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 0.425iT - 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 5.77iT - 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 + 9.78iT - 47T^{2} \) |
| 53 | \( 1 - 1.33iT - 53T^{2} \) |
| 59 | \( 1 + 9.09iT - 59T^{2} \) |
| 61 | \( 1 - 7.73iT - 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 11.0iT - 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 - 8.31T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 3.47iT - 89T^{2} \) |
| 97 | \( 1 + 3.91iT - 97T^{2} \) |
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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
−8.382502953643564639673287627297, −7.31895378262278681169156882765, −6.63556319691692913232396611290, −6.41287788940149487451232381308, −5.19111143532355646945260724761, −4.69045555938635779247608015461, −3.63907150440695491701729633783, −2.79288363494332084703908361167, −1.94044521913591392212255598327, −0.75868989102485802985119355098,
0.35772578663273753146889474415, 2.05481862943536008983122149508, 2.85101564306823886582747022542, 3.64831146088898378063755679824, 4.36662022727237827293873554131, 5.16311306277420149246145676783, 6.11475376058154135282117237219, 6.56040625198869525452375221552, 7.42756195564900926827963331465, 7.987640291102429069543847017643
