| L(s) = 1 | − 1.98·2-s + 2.22·3-s + 1.93·4-s + 0.712·5-s − 4.40·6-s + 4.39·7-s + 0.135·8-s + 1.93·9-s − 1.41·10-s + 3.35·11-s + 4.29·12-s − 2.53·13-s − 8.71·14-s + 1.58·15-s − 4.13·16-s + 4.59·17-s − 3.84·18-s − 3.55·19-s + 1.37·20-s + 9.76·21-s − 6.65·22-s − 2.55·23-s + 0.300·24-s − 4.49·25-s + 5.01·26-s − 2.36·27-s + 8.49·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 1.28·3-s + 0.965·4-s + 0.318·5-s − 1.79·6-s + 1.66·7-s + 0.0477·8-s + 0.645·9-s − 0.446·10-s + 1.01·11-s + 1.23·12-s − 0.702·13-s − 2.32·14-s + 0.408·15-s − 1.03·16-s + 1.11·17-s − 0.905·18-s − 0.815·19-s + 0.307·20-s + 2.13·21-s − 1.41·22-s − 0.532·23-s + 0.0612·24-s − 0.898·25-s + 0.984·26-s − 0.454·27-s + 1.60·28-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
| \(L(1)\) |
\(\approx\) |
\(1.376006099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376006099\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 0.712T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 - 3.73T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 0.655T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 + 4.56T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 9.44T + 71T^{2} \) |
| 73 | \( 1 + 8.94T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 6.59T + 97T^{2} \) |
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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
−10.34530860042422136896650059269, −9.769430559247825469339252230197, −8.841165449154470275595253607810, −8.258683918651663793722728118980, −7.80848502017134205356566041841, −6.75502758931000586844103365416, −5.11250289268051613192387474618, −3.91503383197408593216929668655, −2.25427923826921335556111237064, −1.48338255427089072799554634109,
1.48338255427089072799554634109, 2.25427923826921335556111237064, 3.91503383197408593216929668655, 5.11250289268051613192387474618, 6.75502758931000586844103365416, 7.80848502017134205356566041841, 8.258683918651663793722728118980, 8.841165449154470275595253607810, 9.769430559247825469339252230197, 10.34530860042422136896650059269
