| L(s) = 1 | − 2.27·2-s + 0.433·3-s + 3.17·4-s − 2.29·5-s − 0.986·6-s − 4.86·7-s − 2.67·8-s − 2.81·9-s + 5.22·10-s − 11-s + 1.37·12-s + 0.971·13-s + 11.0·14-s − 0.996·15-s − 0.269·16-s + 2.08·17-s + 6.39·18-s − 6.12·19-s − 7.29·20-s − 2.11·21-s + 2.27·22-s − 7.99·23-s − 1.15·24-s + 0.276·25-s − 2.21·26-s − 2.52·27-s − 15.4·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 0.250·3-s + 1.58·4-s − 1.02·5-s − 0.402·6-s − 1.83·7-s − 0.944·8-s − 0.937·9-s + 1.65·10-s − 0.301·11-s + 0.397·12-s + 0.269·13-s + 2.95·14-s − 0.257·15-s − 0.0674·16-s + 0.505·17-s + 1.50·18-s − 1.40·19-s − 1.63·20-s − 0.460·21-s + 0.484·22-s − 1.66·23-s − 0.236·24-s + 0.0552·25-s − 0.433·26-s − 0.485·27-s − 2.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1060311003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1060311003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 - 0.433T + 3T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 + 4.86T + 7T^{2} \) |
| 13 | \( 1 - 0.971T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 + 7.99T + 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 6.33T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 + 9.85T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 9.60T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.54T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.414498216660583157430607024525, −8.698023402858188035704815908999, −8.138356052912474791522362314166, −7.42575916795219314284224465787, −6.50760062521261834785658929748, −5.87093807506452735317266876241, −4.06646319167901165893323179603, −3.24950870718439651523033607937, −2.22000883927759273266238962916, −0.26932845400060595995433109751,
0.26932845400060595995433109751, 2.22000883927759273266238962916, 3.24950870718439651523033607937, 4.06646319167901165893323179603, 5.87093807506452735317266876241, 6.50760062521261834785658929748, 7.42575916795219314284224465787, 8.138356052912474791522362314166, 8.698023402858188035704815908999, 9.414498216660583157430607024525
