L(s) = 1 | + 2.19·2-s − 2.37·3-s + 2.82·4-s + 5-s − 5.21·6-s − 1.56·7-s + 1.82·8-s + 2.63·9-s + 2.19·10-s − 2.91·11-s − 6.71·12-s + 3.27·13-s − 3.44·14-s − 2.37·15-s − 1.65·16-s − 17-s + 5.78·18-s + 0.0652·19-s + 2.82·20-s + 3.72·21-s − 6.39·22-s + 8.58·23-s − 4.32·24-s + 25-s + 7.19·26-s + 0.872·27-s − 4.43·28-s + ⋯ |
L(s) = 1 | + 1.55·2-s − 1.37·3-s + 1.41·4-s + 0.447·5-s − 2.12·6-s − 0.592·7-s + 0.643·8-s + 0.877·9-s + 0.694·10-s − 0.877·11-s − 1.93·12-s + 0.908·13-s − 0.921·14-s − 0.612·15-s − 0.414·16-s − 0.242·17-s + 1.36·18-s + 0.0149·19-s + 0.632·20-s + 0.812·21-s − 1.36·22-s + 1.79·23-s − 0.882·24-s + 0.200·25-s + 1.41·26-s + 0.167·27-s − 0.838·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 + 2.37T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 19 | \( 1 - 0.0652T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 - 8.02T + 41T^{2} \) |
| 43 | \( 1 + 13.0T + 43T^{2} \) |
| 47 | \( 1 + 0.641T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 73 | \( 1 - 3.08T + 73T^{2} \) |
| 79 | \( 1 - 2.11T + 79T^{2} \) |
| 83 | \( 1 + 6.22T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
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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
−7.18703895293235777515566628226, −6.51886347186819296507992182836, −6.20107323437305219337270918405, −5.38666672982905551913765450449, −5.08400088331957324176616253235, −4.32234909870375438196084070017, −3.29770399523568415446778195630, −2.72742374847866004327714002743, −1.39903618295165023001117896380, 0,
1.39903618295165023001117896380, 2.72742374847866004327714002743, 3.29770399523568415446778195630, 4.32234909870375438196084070017, 5.08400088331957324176616253235, 5.38666672982905551913765450449, 6.20107323437305219337270918405, 6.51886347186819296507992182836, 7.18703895293235777515566628226