| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 6·11-s − 12-s − 3·13-s + 16-s − 17-s − 2·18-s − 7·19-s − 6·22-s + 8·23-s − 24-s − 3·26-s + 5·27-s − 5·29-s + 5·31-s + 32-s + 6·33-s − 34-s − 2·36-s − 8·37-s − 7·38-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 1.60·19-s − 1.27·22-s + 1.66·23-s − 0.204·24-s − 0.588·26-s + 0.962·27-s − 0.928·29-s + 0.898·31-s + 0.176·32-s + 1.04·33-s − 0.171·34-s − 1/3·36-s − 1.31·37-s − 1.13·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−10.09302879555825736775380403270, −8.818483020797254296319231244494, −7.950212259455860680268307957004, −7.00592557180030464461653480323, −6.10743012598324664351263700121, −5.15650183396623031751358571793, −4.72568208639362645794050262121, −3.14111186102510721424588268243, −2.28378669247109526633039618133, 0,
2.28378669247109526633039618133, 3.14111186102510721424588268243, 4.72568208639362645794050262121, 5.15650183396623031751358571793, 6.10743012598324664351263700121, 7.00592557180030464461653480323, 7.950212259455860680268307957004, 8.818483020797254296319231244494, 10.09302879555825736775380403270
