L(s) = 1 | − 2.82·3-s − 2·7-s + 5.00·9-s − 11-s + 1.17·13-s − 6.82·17-s + 5.65·21-s − 2.82·23-s − 5.65·27-s − 3.65·29-s + 2.82·33-s + 7.65·37-s − 3.31·39-s + 6·41-s − 6·43-s + 2.82·47-s − 3·49-s + 19.3·51-s − 11.6·53-s − 1.65·59-s − 9.31·61-s − 10.0·63-s + 12.4·67-s + 8.00·69-s − 11.3·71-s + 1.17·73-s + 2·77-s + ⋯ |
L(s) = 1 | − 1.63·3-s − 0.755·7-s + 1.66·9-s − 0.301·11-s + 0.324·13-s − 1.65·17-s + 1.23·21-s − 0.589·23-s − 1.08·27-s − 0.679·29-s + 0.492·33-s + 1.25·37-s − 0.530·39-s + 0.937·41-s − 0.914·43-s + 0.412·47-s − 0.428·49-s + 2.70·51-s − 1.60·53-s − 0.215·59-s − 1.19·61-s − 1.25·63-s + 1.52·67-s + 0.963·69-s − 1.34·71-s + 0.137·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4234479867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4234479867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 1.65T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 1.17T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−8.307323129530721521352898678046, −7.39668417147985976957974068277, −6.62129709077873261109247372357, −6.19196640897290924996541778752, −5.59080505299226006923794607029, −4.65178540046334925296204728728, −4.12401290521254449406907223166, −2.92045989125510829866317990895, −1.72686802855194500088937955603, −0.38995547894975355180916965397,
0.38995547894975355180916965397, 1.72686802855194500088937955603, 2.92045989125510829866317990895, 4.12401290521254449406907223166, 4.65178540046334925296204728728, 5.59080505299226006923794607029, 6.19196640897290924996541778752, 6.62129709077873261109247372357, 7.39668417147985976957974068277, 8.307323129530721521352898678046