| L(s) = 1 | + 2-s + 2.21·3-s + 4-s + 2.54·5-s + 2.21·6-s + 7-s + 8-s + 1.88·9-s + 2.54·10-s − 1.12·11-s + 2.21·12-s + 4.15·13-s + 14-s + 5.63·15-s + 16-s + 2.25·17-s + 1.88·18-s − 2.89·19-s + 2.54·20-s + 2.21·21-s − 1.12·22-s + 4.35·23-s + 2.21·24-s + 1.49·25-s + 4.15·26-s − 2.45·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.27·3-s + 0.5·4-s + 1.14·5-s + 0.902·6-s + 0.377·7-s + 0.353·8-s + 0.629·9-s + 0.806·10-s − 0.339·11-s + 0.638·12-s + 1.15·13-s + 0.267·14-s + 1.45·15-s + 0.250·16-s + 0.547·17-s + 0.445·18-s − 0.663·19-s + 0.570·20-s + 0.482·21-s − 0.239·22-s + 0.907·23-s + 0.451·24-s + 0.299·25-s + 0.815·26-s − 0.472·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.995667176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.995667176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 2.54T + 5T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 - 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.25T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 4.35T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 + 8.52T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 - 1.76T + 47T^{2} \) |
| 53 | \( 1 + 9.58T + 53T^{2} \) |
| 59 | \( 1 - 6.05T + 59T^{2} \) |
| 61 | \( 1 + 7.72T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 + 8.03T + 71T^{2} \) |
| 73 | \( 1 - 4.31T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−8.196403673596257808872716391042, −7.38152068825806497010001024346, −6.57637348487527818588957836850, −5.85170455616762285131669719325, −5.25741714136672923493024654623, −4.34122874504989494642122287768, −3.47215396083505014171743004882, −2.85511566969113389500555334838, −2.04818847872679856263830120370, −1.35387221013856638499373080374,
1.35387221013856638499373080374, 2.04818847872679856263830120370, 2.85511566969113389500555334838, 3.47215396083505014171743004882, 4.34122874504989494642122287768, 5.25741714136672923493024654623, 5.85170455616762285131669719325, 6.57637348487527818588957836850, 7.38152068825806497010001024346, 8.196403673596257808872716391042
