| L(s) = 1 | + 1.73i·5-s + 3.61·7-s + 0.734i·11-s + 0.609i·13-s + 5.52·17-s − 2i·19-s + 4.73·23-s + 2.00·25-s − 7.83i·29-s − 9.41·31-s + 6.25i·35-s − 2.34i·37-s + 8.52·41-s − 10.2i·43-s + 11.7·47-s + ⋯ |
| L(s) = 1 | + 0.774i·5-s + 1.36·7-s + 0.221i·11-s + 0.169i·13-s + 1.33·17-s − 0.458i·19-s + 0.987·23-s + 0.400·25-s − 1.45i·29-s − 1.69·31-s + 1.05i·35-s − 0.384i·37-s + 1.33·41-s − 1.56i·43-s + 1.71·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.632663059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.632663059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 0.734iT - 11T^{2} \) |
| 13 | \( 1 - 0.609iT - 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 7.83iT - 29T^{2} \) |
| 31 | \( 1 + 9.41T + 31T^{2} \) |
| 37 | \( 1 + 2.34iT - 37T^{2} \) |
| 41 | \( 1 - 8.52T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 13.0iT - 53T^{2} \) |
| 59 | \( 1 + 1.20iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 6.31iT - 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 - 2.49T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 1.94T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.098757154731334963379010785861, −7.36710573610739514473693460488, −7.16197146145094541306381541861, −5.92621351592753190205941737443, −5.41174445631963234512796427054, −4.54970648829569612454877848409, −3.79172045498339632596712959612, −2.78856211513326535314365198184, −1.99950505746416678884163359544, −0.909925093846414396776528693836,
1.02437140318962088342965296932, 1.53724134964069856032267174280, 2.81679148557306438096113187637, 3.75908749073140880305992424836, 4.64276489632837592105105033285, 5.32713029324441317400327208148, 5.65228525521903227030158106349, 6.91173785816293684234491364539, 7.63655374575917053762182000808, 8.136937706011489603896750431262
