| L(s) = 1 | − 0.837·2-s + 3-s − 1.29·4-s + 2.51·5-s − 0.837·6-s + 2.76·8-s + 9-s − 2.10·10-s + 3.06·11-s − 1.29·12-s − 3.98·13-s + 2.51·15-s + 0.286·16-s + 4.96·17-s − 0.837·18-s + 5.49·19-s − 3.26·20-s − 2.56·22-s + 2.97·23-s + 2.76·24-s + 1.32·25-s + 3.33·26-s + 27-s − 1.60·29-s − 2.10·30-s + 7.48·31-s − 5.76·32-s + ⋯ |
| L(s) = 1 | − 0.591·2-s + 0.577·3-s − 0.649·4-s + 1.12·5-s − 0.341·6-s + 0.976·8-s + 0.333·9-s − 0.665·10-s + 0.923·11-s − 0.375·12-s − 1.10·13-s + 0.649·15-s + 0.0717·16-s + 1.20·17-s − 0.197·18-s + 1.26·19-s − 0.730·20-s − 0.546·22-s + 0.619·23-s + 0.563·24-s + 0.264·25-s + 0.654·26-s + 0.192·27-s − 0.298·29-s − 0.384·30-s + 1.34·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.266728875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266728875\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.837T + 2T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + 3.98T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 5.49T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 + 1.60T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 43 | \( 1 - 0.973T + 43T^{2} \) |
| 47 | \( 1 - 0.697T + 47T^{2} \) |
| 53 | \( 1 + 6.26T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 0.442T + 61T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 + 0.791T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 5.22T + 97T^{2} \) |
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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
−8.041803482953362539514294236168, −7.61864893904082802479426422878, −6.82385948226384594788389183051, −5.91543867628495937982628414514, −5.16122151286906497676461716004, −4.54252907478360437473092569881, −3.51751328185500985004351337010, −2.71255604710847038909480293500, −1.62922203806142096786816825473, −0.933789212673705436567896640513,
0.933789212673705436567896640513, 1.62922203806142096786816825473, 2.71255604710847038909480293500, 3.51751328185500985004351337010, 4.54252907478360437473092569881, 5.16122151286906497676461716004, 5.91543867628495937982628414514, 6.82385948226384594788389183051, 7.61864893904082802479426422878, 8.041803482953362539514294236168
