| L(s) = 1 | − 7.15·7-s + 5.06i·11-s + 3.12·13-s − 8.72i·17-s + 20.1·19-s − 14.7i·23-s + 39.7i·29-s − 39.3·31-s − 34.8·37-s − 13.2i·41-s + 66.6·43-s + 16.9i·47-s + 2.18·49-s − 4.62i·53-s + 25.7i·59-s + ⋯ |
| L(s) = 1 | − 1.02·7-s + 0.460i·11-s + 0.240·13-s − 0.513i·17-s + 1.06·19-s − 0.640i·23-s + 1.37i·29-s − 1.27·31-s − 0.942·37-s − 0.323i·41-s + 1.54·43-s + 0.359i·47-s + 0.0445·49-s − 0.0873i·53-s + 0.436i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.065067354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065067354\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7.15T + 49T^{2} \) |
| 11 | \( 1 - 5.06iT - 121T^{2} \) |
| 13 | \( 1 - 3.12T + 169T^{2} \) |
| 17 | \( 1 + 8.72iT - 289T^{2} \) |
| 19 | \( 1 - 20.1T + 361T^{2} \) |
| 23 | \( 1 + 14.7iT - 529T^{2} \) |
| 29 | \( 1 - 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 39.3T + 961T^{2} \) |
| 37 | \( 1 + 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 66.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 4.62iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 25.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 106.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 23.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 66.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 154. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 175.T + 9.40e3T^{2} \) |
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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
−8.708998240726924756604261309501, −7.37809717075041888596850145852, −7.17212408899163480649583958098, −6.15292004651090441203751021510, −5.43514934532452236371686198713, −4.52636604765844613678453560514, −3.49139664574763532968192392191, −2.85821292579746878124116397397, −1.60259326198301978677811521371, −0.29472335673806291694807542070,
0.931021552056745294780123316688, 2.23027330545164702421569079494, 3.36023176918669929384655724894, 3.80439333566493039030870099879, 5.06816852565246859853560033522, 5.89186355691402580983322896436, 6.43131279299327212312906004501, 7.41319655171403216542803211092, 7.980923059344348806781883849728, 9.095804991438078334135959343604
