| L(s) = 1 | + 2·3-s + 7-s + 9-s − 6·13-s + 2·21-s + 23-s − 5·25-s − 4·27-s − 2·29-s + 2·31-s + 2·37-s − 12·39-s + 6·41-s − 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s − 12·61-s + 63-s − 4·67-s + 2·69-s + 2·73-s − 10·75-s + 8·79-s − 11·81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s − 0.371·29-s + 0.359·31-s + 0.328·37-s − 1.92·39-s + 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s − 1.53·61-s + 0.125·63-s − 0.488·67-s + 0.240·69-s + 0.234·73-s − 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.53884486751152, −13.28351230412339, −12.58283389484126, −12.16734717291155, −11.65959567859652, −11.27519157576202, −10.57949223125380, −9.990254897517034, −9.699130826689340, −9.207220407151156, −8.741877859536714, −8.111206197426220, −7.889279174315548, −7.199466130286387, −7.058219195834685, −6.113468113165652, −5.575594505114041, −5.091803339844962, −4.388626196459258, −4.029574157405318, −3.286948299204818, −2.770676733146966, −2.204534551004625, −1.891905770610657, −0.8681987501599210, 0,
0.8681987501599210, 1.891905770610657, 2.204534551004625, 2.770676733146966, 3.286948299204818, 4.029574157405318, 4.388626196459258, 5.091803339844962, 5.575594505114041, 6.113468113165652, 7.058219195834685, 7.199466130286387, 7.889279174315548, 8.111206197426220, 8.741877859536714, 9.207220407151156, 9.699130826689340, 9.990254897517034, 10.57949223125380, 11.27519157576202, 11.65959567859652, 12.16734717291155, 12.58283389484126, 13.28351230412339, 13.53884486751152
