| L(s) = 1 | + 0.0847·2-s − 1.52·3-s − 1.99·4-s + 2.94·5-s − 0.129·6-s − 4.64·7-s − 0.338·8-s − 0.664·9-s + 0.249·10-s + 2.78·11-s + 3.04·12-s + 5.35·13-s − 0.394·14-s − 4.49·15-s + 3.95·16-s + 1.40·17-s − 0.0563·18-s − 2.15·19-s − 5.86·20-s + 7.10·21-s + 0.235·22-s + 2.38·23-s + 0.517·24-s + 3.65·25-s + 0.453·26-s + 5.60·27-s + 9.26·28-s + ⋯ |
| L(s) = 1 | + 0.0599·2-s − 0.882·3-s − 0.996·4-s + 1.31·5-s − 0.0528·6-s − 1.75·7-s − 0.119·8-s − 0.221·9-s + 0.0788·10-s + 0.838·11-s + 0.879·12-s + 1.48·13-s − 0.105·14-s − 1.16·15-s + 0.989·16-s + 0.341·17-s − 0.0132·18-s − 0.495·19-s − 1.31·20-s + 1.55·21-s + 0.0502·22-s + 0.498·23-s + 0.105·24-s + 0.730·25-s + 0.0890·26-s + 1.07·27-s + 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9271943033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9271943033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0847T + 2T^{2} \) |
| 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 7.29T + 37T^{2} \) |
| 41 | \( 1 - 2.27T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 9.60T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.64T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 8.64T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−10.59137373283937815612705114164, −9.832149288778143994280670614531, −9.264434577956598094599871444738, −8.486304273337451167031883253290, −6.58652307110004852676472794196, −6.12824349517948985459020146830, −5.56334488959119658647645265344, −4.15208010007429650397592459271, −3.02842297951875637879611009272, −0.921481936145754945716415602537,
0.921481936145754945716415602537, 3.02842297951875637879611009272, 4.15208010007429650397592459271, 5.56334488959119658647645265344, 6.12824349517948985459020146830, 6.58652307110004852676472794196, 8.486304273337451167031883253290, 9.264434577956598094599871444738, 9.832149288778143994280670614531, 10.59137373283937815612705114164
