| L(s) = 1 | + 1.17·3-s + 2.23i·5-s + 2.64i·7-s − 1.62·9-s − 0.359·11-s + 5.64i·13-s + 2.62i·15-s − 2.03·17-s + 3.10i·21-s − 5.00·25-s − 5.42·27-s − 10.7i·29-s − 0.422·33-s − 5.91·35-s + 6.62i·39-s + ⋯ |
| L(s) = 1 | + 0.677·3-s + 0.999i·5-s + 0.999i·7-s − 0.541·9-s − 0.108·11-s + 1.56i·13-s + 0.677i·15-s − 0.492·17-s + 0.677i·21-s − 1.00·25-s − 1.04·27-s − 1.99i·29-s − 0.0735·33-s − 0.999·35-s + 1.06i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.230302557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230302557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 11 | \( 1 + 0.359T + 11T^{2} \) |
| 13 | \( 1 - 5.64iT - 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.7iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 5.71iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.346386180093342466273748004146, −8.630847796357955858846123142120, −7.980744372098043971933664559086, −7.03521097611726556240738905274, −6.30859909160481128116856847992, −5.64321322153395250656025913285, −4.40355984238123731316216426941, −3.53084062958988069730206883418, −2.46447419528241520702722251734, −2.12370449815438032970853931703,
0.36131908307115865791224792721, 1.55750273770410683843923941433, 2.93487043838718096771969679922, 3.62458372833136434131846735209, 4.69006961760661883586383706611, 5.36689992082535176608101740300, 6.29634304877482370307338280368, 7.50559764282242654939611497619, 7.889869808557509593655856404227, 8.740061325548834873367079011758
