| L(s) = 1 | + 2.23i·7-s + 4.47·11-s + i·13-s + 2i·17-s + 6.70·19-s + 4.47i·23-s + 6·29-s − 2.23·31-s − 2i·37-s − 2·41-s + 6.70i·43-s − 8.94i·47-s + 1.99·49-s + 6i·53-s + 8.94·59-s + ⋯ |
| L(s) = 1 | + 0.845i·7-s + 1.34·11-s + 0.277i·13-s + 0.485i·17-s + 1.53·19-s + 0.932i·23-s + 1.11·29-s − 0.401·31-s − 0.328i·37-s − 0.312·41-s + 1.02i·43-s − 1.30i·47-s + 0.285·49-s + 0.824i·53-s + 1.16·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.363443248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.363443248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.23iT - 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.70iT - 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 5T + 61T^{2} \) |
| 67 | \( 1 - 2.23iT - 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 8.94iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−8.072438087572893142118973671112, −7.29631941575338722748906364510, −6.65131385898945773837346405462, −5.90328160373095177621867233790, −5.35766988516932201337318361694, −4.45741976984978875646495786072, −3.64381053108762306175143022369, −2.95403865548402977492300796172, −1.86437307009942972937616188467, −1.07949151596304103995789279312,
0.68982613261272262239603592805, 1.38731723931150300749994337858, 2.66880491508167468515853968243, 3.48835510292969282330288976286, 4.16758225957190676078747827331, 4.89836723953848653239487408142, 5.69945972240431362658351775998, 6.62280360700136194916089388955, 7.00319762837360224856736714358, 7.72286727691192280822908932675
