| L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 6·17-s + 19-s − 4·23-s − 25-s + 27-s + 2·29-s + 4·31-s + 4·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s − 2·45-s − 4·47-s − 7·49-s − 6·51-s + 10·53-s − 8·55-s + 57-s − 12·59-s − 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.696·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.583·47-s − 49-s − 0.840·51-s + 1.37·53-s − 1.07·55-s + 0.132·57-s − 1.56·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.219893553992030605447454618201, −7.48311886504808907872736816754, −6.79642733627755592406769031534, −6.14228543534174242078427935073, −4.84503850255708974062842753376, −4.20573906208166268740620008576, −3.58691114893739395424605483185, −2.56070554639047194803817756680, −1.53339445883331929692810845376, 0,
1.53339445883331929692810845376, 2.56070554639047194803817756680, 3.58691114893739395424605483185, 4.20573906208166268740620008576, 4.84503850255708974062842753376, 6.14228543534174242078427935073, 6.79642733627755592406769031534, 7.48311886504808907872736816754, 8.219893553992030605447454618201
