| L(s) = 1 | − 2.64i·7-s − 4i·11-s + 8i·23-s + 5·25-s + 10.5·29-s + 6·37-s − 5.29i·43-s − 7.00·49-s − 10.5·53-s − 15.8i·67-s − 16i·71-s − 10.5·77-s − 15.8i·79-s + 20i·107-s − 18·109-s + ⋯ |
| L(s) = 1 | − 0.999i·7-s − 1.20i·11-s + 1.66i·23-s + 25-s + 1.96·29-s + 0.986·37-s − 0.806i·43-s − 49-s − 1.45·53-s − 1.93i·67-s − 1.89i·71-s − 1.20·77-s − 1.78i·79-s + 1.93i·107-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730145568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730145568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.082886458889826496594277872760, −7.69217505743594004603036149725, −6.69613963722271902006333571700, −6.20146915761756518609893768712, −5.20333960665526781470230793225, −4.49161258302568380254285976125, −3.49865007369641657847459771732, −2.97349728459016405634883908921, −1.49753579250095489242734955624, −0.55930821768041445682052165240,
1.16805508443323368406741971621, 2.45299790766402897617289449363, 2.84521111615296403019927292877, 4.34332896189326234250723147267, 4.73659766889800966547832760955, 5.65662771431780334591192673390, 6.56978980148214559612966411250, 6.94487666453444370642340909288, 8.172604640953835828351816884376, 8.447423291585331862011369011331
