| L(s) = 1 | + 4i·2-s − 16·4-s − 176i·7-s − 64i·8-s + 60·11-s − 658i·13-s + 704·14-s + 256·16-s − 414i·17-s − 956·19-s + 240i·22-s − 600i·23-s + 2.63e3·26-s + 2.81e3i·28-s + 5.57e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.35i·7-s − 0.353i·8-s + 0.149·11-s − 1.07i·13-s + 0.959·14-s + 0.250·16-s − 0.347i·17-s − 0.607·19-s + 0.105i·22-s − 0.236i·23-s + 0.763·26-s + 0.678i·28-s + 1.23·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4058504671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4058504671\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 176iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 60T + 1.61e5T^{2} \) |
| 13 | \( 1 + 658iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 414iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 956T + 2.47e6T^{2} \) |
| 23 | \( 1 + 600iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.45e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.91e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.33e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.96e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.12e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.68e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.55e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.46e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.66e5iT - 8.58e9T^{2} \) |
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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
−10.11331200178994557150058408981, −8.776506177212894162431892647298, −7.954729427233739024662154568846, −7.11941035391945877398894704064, −6.32969307137263377534888027910, −5.10288064414031040979642756947, −4.19554846174002399751199244839, −3.07925598898133590186809360902, −1.18395069004212450496864571581, −0.10134902057674442990477262355,
1.63678104265634774763270944253, 2.47040574690854360132459868575, 3.72296340769228336339451732452, 4.85438675428714799650903514674, 5.86716610419857470424684764600, 6.88333211029644580372838160426, 8.361062556335357317445858470030, 8.916212591585891512448789302863, 9.736340261899489014752228778938, 10.76338165471746624580506418690
