| L(s) = 1 | + i·3-s − i·5-s − 3.12·7-s − 9-s + 1.96·11-s − 2.52·13-s + 15-s + 3.66i·17-s + 6.78·19-s − 3.12i·21-s + (−2.20 − 4.25i)23-s − 25-s − i·27-s + 2.23·29-s − 0.377i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 1.18·7-s − 0.333·9-s + 0.593·11-s − 0.699·13-s + 0.258·15-s + 0.887i·17-s + 1.55·19-s − 0.681i·21-s + (−0.460 − 0.887i)23-s − 0.200·25-s − 0.192i·27-s + 0.415·29-s − 0.0678i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4352709334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4352709334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.20 + 4.25i)T \) |
| good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 1.96T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 - 3.66iT - 17T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 + 0.377iT - 31T^{2} \) |
| 37 | \( 1 - 5.26iT - 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 1.35iT - 47T^{2} \) |
| 53 | \( 1 + 3.17iT - 53T^{2} \) |
| 59 | \( 1 - 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 4.62iT - 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 15.1iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 0.338iT - 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−8.015072213554432296053818269593, −7.12690793635783053698203927195, −6.40479082870677694004125826450, −5.78805608658534442823598579177, −4.92926686922646694626877500259, −4.21761675654884693655501996319, −3.40606407041544983292129560193, −2.75428151198636461212228819004, −1.43566245698884529292872669316, −0.12445651061338409620162116750,
1.09777882119925702496653052191, 2.31690319055976191668844054856, 3.13641658983627061779137405669, 3.65713655682032938425316571371, 4.88839433219908545518486039503, 5.65671027690902460549553466504, 6.38547961762846632351839028348, 7.06372712186213335345248284241, 7.41781782185572770605889155352, 8.265408465957288579816419604746
