| L(s) = 1 | + i·3-s + (−2.21 + 0.311i)5-s − 1.70i·7-s − 9-s − 3.78·11-s + 1.17i·13-s + (−0.311 − 2.21i)15-s − 4.26i·17-s + 6.98·19-s + 1.70·21-s − i·23-s + (4.80 − 1.37i)25-s − i·27-s + 4.26·29-s − 0.0889·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.990 + 0.139i)5-s − 0.645i·7-s − 0.333·9-s − 1.13·11-s + 0.326i·13-s + (−0.0803 − 0.571i)15-s − 1.03i·17-s + 1.60·19-s + 0.372·21-s − 0.208i·23-s + (0.961 − 0.275i)25-s − 0.192i·27-s + 0.791·29-s − 0.0159·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9236163142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9236163142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.21 - 0.311i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.70iT - 7T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 - 1.17iT - 13T^{2} \) |
| 17 | \( 1 + 4.26iT - 17T^{2} \) |
| 19 | \( 1 - 6.98T + 19T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 0.0889T + 31T^{2} \) |
| 37 | \( 1 - 1.72iT - 37T^{2} \) |
| 41 | \( 1 + 7.69T + 41T^{2} \) |
| 43 | \( 1 - 8.76iT - 43T^{2} \) |
| 47 | \( 1 - 2.25iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 1.64T + 71T^{2} \) |
| 73 | \( 1 - 9.70iT - 73T^{2} \) |
| 79 | \( 1 + 6.07T + 79T^{2} \) |
| 83 | \( 1 + 6.53iT - 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 - 7.78iT - 97T^{2} \) |
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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
−9.054271927231617486251805009040, −8.141302558860686787748548965159, −7.52601372551930374120460462824, −7.01461330834573007759076247580, −5.84618117278459218844888224305, −4.82402454135454555492141289263, −4.47961254574421638056389936534, −3.27467085077401159372092815583, −2.82545293861900877192827146944, −0.954924861938734932505303092173,
0.37542856551273782180728761334, 1.77261883877302573789279821945, 2.95243100732242521068029484210, 3.58039254295701045760794482891, 4.89756042072699770157475072633, 5.42458281058337351593447641918, 6.35038524467416229887158870808, 7.32650849411642804540332171898, 7.84512214713871391279428461486, 8.422335589724333586995872027867
