L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 3·5-s − 2·6-s − 7-s + 9-s − 6·10-s − 2·12-s + 2·13-s − 2·14-s + 3·15-s − 4·16-s + 3·17-s + 2·18-s − 4·19-s − 6·20-s + 21-s + 2·23-s + 4·25-s + 4·26-s − 27-s − 2·28-s + 8·29-s + 6·30-s + 2·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 1.89·10-s − 0.577·12-s + 0.554·13-s − 0.534·14-s + 0.774·15-s − 16-s + 0.727·17-s + 0.471·18-s − 0.917·19-s − 1.34·20-s + 0.218·21-s + 0.417·23-s + 4/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.48·29-s + 1.09·30-s + 0.359·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.097588589\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.097588589\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.761942686918452707171803858900, −7.947566965552207654804548696847, −7.14335200016652884466702769572, −6.29742246849264780562506493129, −5.81561730371896548427627817680, −4.66708477306658625952972274470, −4.27239898558423069954385134608, −3.48069358835781656468202953772, −2.64368557607058645503220519227, −0.75458677265401217618730495619,
0.75458677265401217618730495619, 2.64368557607058645503220519227, 3.48069358835781656468202953772, 4.27239898558423069954385134608, 4.66708477306658625952972274470, 5.81561730371896548427627817680, 6.29742246849264780562506493129, 7.14335200016652884466702769572, 7.947566965552207654804548696847, 8.761942686918452707171803858900