| L(s) = 1 | + (−1.68 + 1.68i)5-s + 7-s + (−1.15 − 1.15i)11-s + (−1.23 + 1.23i)13-s − 0.707i·17-s + (5.82 + 5.82i)19-s − 4.09i·23-s − 0.700i·25-s + (−1.57 − 1.57i)29-s + 6.27i·31-s + (−1.68 + 1.68i)35-s + (−4.88 − 4.88i)37-s + 3.51·41-s + (−4.39 + 4.39i)43-s + 1.52·47-s + ⋯ |
| L(s) = 1 | + (−0.755 + 0.755i)5-s + 0.377·7-s + (−0.347 − 0.347i)11-s + (−0.342 + 0.342i)13-s − 0.171i·17-s + (1.33 + 1.33i)19-s − 0.854i·23-s − 0.140i·25-s + (−0.292 − 0.292i)29-s + 1.12i·31-s + (−0.285 + 0.285i)35-s + (−0.803 − 0.803i)37-s + 0.548·41-s + (−0.670 + 0.670i)43-s + 0.222·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7429854866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7429854866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + (1.68 - 1.68i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.15 + 1.15i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.23 - 1.23i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.707iT - 17T^{2} \) |
| 19 | \( 1 + (-5.82 - 5.82i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.09iT - 23T^{2} \) |
| 29 | \( 1 + (1.57 + 1.57i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.27iT - 31T^{2} \) |
| 37 | \( 1 + (4.88 + 4.88i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.51T + 41T^{2} \) |
| 43 | \( 1 + (4.39 - 4.39i)T - 43iT^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + (-2.98 + 2.98i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.0546 + 0.0546i)T + 59iT^{2} \) |
| 61 | \( 1 + (8.07 - 8.07i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.27 - 9.27i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.21iT - 71T^{2} \) |
| 73 | \( 1 + 0.995iT - 73T^{2} \) |
| 79 | \( 1 + 6.85iT - 79T^{2} \) |
| 83 | \( 1 + (7.92 - 7.92i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 7.84T + 97T^{2} \) |
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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
−8.611161712166130539483278784544, −7.911517558585079781129997508181, −7.36669836463781280377088328267, −6.73876410502047218778004071683, −5.74387072693171640165867831938, −5.07324865440334594090164751616, −4.06322036533585338112209887733, −3.37576853262604462282642968773, −2.55885837477541247928131888735, −1.32277252853388820173404179182,
0.22790110116110443912209628982, 1.35527003090376988749941247625, 2.57993597823190425474946553779, 3.52732782687330014427826937575, 4.44375848022263224598008765965, 5.07556504183384982940259435870, 5.64587774063419548837712321941, 6.90528054114500140401590356807, 7.49058907794566459635151778407, 8.070240585715301591686967636355
