| L(s) = 1 | − 2-s − 2.58·3-s + 4-s − 2.90·5-s + 2.58·6-s − 7-s − 8-s + 3.65·9-s + 2.90·10-s + 1.98·11-s − 2.58·12-s − 1.13·13-s + 14-s + 7.49·15-s + 16-s + 6.51·17-s − 3.65·18-s + 6.40·19-s − 2.90·20-s + 2.58·21-s − 1.98·22-s + 2.52·23-s + 2.58·24-s + 3.42·25-s + 1.13·26-s − 1.70·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s − 1.29·5-s + 1.05·6-s − 0.377·7-s − 0.353·8-s + 1.21·9-s + 0.917·10-s + 0.597·11-s − 0.744·12-s − 0.315·13-s + 0.267·14-s + 1.93·15-s + 0.250·16-s + 1.58·17-s − 0.862·18-s + 1.46·19-s − 0.649·20-s + 0.563·21-s − 0.422·22-s + 0.527·23-s + 0.526·24-s + 0.685·25-s + 0.222·26-s − 0.327·27-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
| \(L(1)\) |
\(\approx\) |
\(0.6538590473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6538590473\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 - 6.40T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 0.614T + 43T^{2} \) |
| 47 | \( 1 - 6.59T + 47T^{2} \) |
| 53 | \( 1 - 4.77T + 53T^{2} \) |
| 59 | \( 1 + 4.80T + 59T^{2} \) |
| 61 | \( 1 - 5.92T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.79T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 0.899T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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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
−7.83644439361281844002775753946, −7.39561016506271007300760164368, −6.78895258954011237492512465846, −5.98576462997948841931469835114, −5.35617852426957797980580249134, −4.54797504338195638238710801433, −3.64157741286351606633163316408, −2.89484327729758742916168707671, −1.13959688450968355724241334381, −0.64015562112288881359343420960,
0.64015562112288881359343420960, 1.13959688450968355724241334381, 2.89484327729758742916168707671, 3.64157741286351606633163316408, 4.54797504338195638238710801433, 5.35617852426957797980580249134, 5.98576462997948841931469835114, 6.78895258954011237492512465846, 7.39561016506271007300760164368, 7.83644439361281844002775753946
