| L(s) = 1 | − i·2-s + 0.0670i·3-s − 4-s + (−1.65 + 1.50i)5-s + 0.0670·6-s + 5.03i·7-s + i·8-s + 2.99·9-s + (1.50 + 1.65i)10-s − 3.28·11-s − 0.0670i·12-s + 5.03·14-s + (−0.101 − 0.110i)15-s + 16-s − 0.516i·17-s − 2.99i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.0386i·3-s − 0.5·4-s + (−0.738 + 0.674i)5-s + 0.0273·6-s + 1.90i·7-s + 0.353i·8-s + 0.998·9-s + (0.476 + 0.522i)10-s − 0.991·11-s − 0.0193i·12-s + 1.34·14-s + (−0.0260 − 0.0285i)15-s + 0.250·16-s − 0.125i·17-s − 0.706i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7172990927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7172990927\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (1.65 - 1.50i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.0670iT - 3T^{2} \) |
| 7 | \( 1 - 5.03iT - 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 17 | \( 1 + 0.516iT - 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 0.524iT - 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 + 6.84T + 31T^{2} \) |
| 37 | \( 1 - 5.49iT - 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 - 9.35iT - 43T^{2} \) |
| 47 | \( 1 - 1.65iT - 47T^{2} \) |
| 53 | \( 1 + 9.30iT - 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 3.38iT - 67T^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 + 6.43iT - 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 2.57iT - 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 1.88iT - 97T^{2} \) |
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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
−9.539841587830957232255777258596, −9.139279384873842703800872913779, −7.934545807310635246551547557741, −7.58414963400639043740209123322, −6.32826923306514192678807703657, −5.40730980055003971674506923645, −4.68287931947849662507980954891, −3.41893925647404024781813921159, −2.78441132369346360489630765688, −1.76948934922001609071441447403,
0.28687897157419935748800358349, 1.39834054928661050692309143943, 3.49874462385033021082518190096, 4.12557542068770293591089274771, 4.83704037163358778599848692741, 5.73328675208153887215535693684, 7.15104477658323142789079110480, 7.46084866291118532889009856822, 7.79034346812264655676948681796, 9.023543338321743643057238738715
