| L(s) = 1 | − 5-s + 0.121·7-s − 2.87·11-s + 5.22·13-s + 2.22·17-s + 1.22·19-s − 23-s + 25-s − 9.34·29-s − 2.12·31-s − 0.121·35-s + 5.59·37-s − 8.22·41-s + 8·43-s + 10.4·47-s − 6.98·49-s + 3.59·53-s + 2.87·55-s + 0.650·59-s − 7.33·61-s − 5.22·65-s + 5.59·67-s + 13.9·71-s + 12.9·73-s − 0.349·77-s − 3.51·79-s + 11.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.0459·7-s − 0.867·11-s + 1.45·13-s + 0.540·17-s + 0.281·19-s − 0.208·23-s + 0.200·25-s − 1.73·29-s − 0.381·31-s − 0.0205·35-s + 0.919·37-s − 1.28·41-s + 1.21·43-s + 1.52·47-s − 0.997·49-s + 0.493·53-s + 0.388·55-s + 0.0846·59-s − 0.939·61-s − 0.648·65-s + 0.683·67-s + 1.65·71-s + 1.51·73-s − 0.0398·77-s − 0.395·79-s + 1.21·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730249888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730249888\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 0.121T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 2.12T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 3.59T + 53T^{2} \) |
| 59 | \( 1 - 0.650T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 5.59T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 3.51T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 0.486T + 89T^{2} \) |
| 97 | \( 1 - 0.635T + 97T^{2} \) |
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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
−7.933941265653865905993035050915, −7.22639220921320099555227471987, −6.41010998119142004645101667310, −5.63013345753158074281748672730, −5.19332366885526801019576690191, −4.04440243182723350194702502142, −3.64142121406332749149034107718, −2.73133099074177703330270970874, −1.71872603270270747772551083490, −0.65559402291503074044996035287,
0.65559402291503074044996035287, 1.71872603270270747772551083490, 2.73133099074177703330270970874, 3.64142121406332749149034107718, 4.04440243182723350194702502142, 5.19332366885526801019576690191, 5.63013345753158074281748672730, 6.41010998119142004645101667310, 7.22639220921320099555227471987, 7.933941265653865905993035050915
