| L(s) = 1 | + 2.54·2-s + 0.588·3-s + 4.46·4-s − 0.268·5-s + 1.49·6-s + 2.27·7-s + 6.27·8-s − 2.65·9-s − 0.682·10-s − 11-s + 2.62·12-s + 2.39·13-s + 5.78·14-s − 0.157·15-s + 7.03·16-s − 6.76·17-s − 6.75·18-s + 3.70·19-s − 1.19·20-s + 1.33·21-s − 2.54·22-s − 8.19·23-s + 3.69·24-s − 4.92·25-s + 6.08·26-s − 3.32·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 0.339·3-s + 2.23·4-s − 0.119·5-s + 0.610·6-s + 0.859·7-s + 2.21·8-s − 0.884·9-s − 0.215·10-s − 0.301·11-s + 0.758·12-s + 0.663·13-s + 1.54·14-s − 0.0407·15-s + 1.75·16-s − 1.64·17-s − 1.59·18-s + 0.851·19-s − 0.268·20-s + 0.291·21-s − 0.542·22-s − 1.70·23-s + 0.753·24-s − 0.985·25-s + 1.19·26-s − 0.639·27-s + 1.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.572923102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.572923102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 - 0.588T + 3T^{2} \) |
| 5 | \( 1 + 0.268T + 5T^{2} \) |
| 7 | \( 1 - 2.27T + 7T^{2} \) |
| 13 | \( 1 - 2.39T + 13T^{2} \) |
| 17 | \( 1 + 6.76T + 17T^{2} \) |
| 19 | \( 1 - 3.70T + 19T^{2} \) |
| 23 | \( 1 + 8.19T + 23T^{2} \) |
| 29 | \( 1 - 6.34T + 29T^{2} \) |
| 31 | \( 1 - 8.61T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 2.20T + 41T^{2} \) |
| 43 | \( 1 - 5.10T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 67 | \( 1 - 0.912T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 7.24T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−11.14114094357935471396006412024, −9.843464560572832750877023546581, −8.386492813572041978805718190988, −7.88219818540643984747098879817, −6.50860835953454163888536539936, −5.90002143820940951805952288675, −4.82387542463935800917711764772, −4.12982228177810684642776964172, −2.97709830153808582187584029217, −2.03307240598656918544256442154,
2.03307240598656918544256442154, 2.97709830153808582187584029217, 4.12982228177810684642776964172, 4.82387542463935800917711764772, 5.90002143820940951805952288675, 6.50860835953454163888536539936, 7.88219818540643984747098879817, 8.386492813572041978805718190988, 9.843464560572832750877023546581, 11.14114094357935471396006412024
