L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 6·11-s + 12-s + 2·13-s + 2·14-s + 16-s + 18-s + 2·21-s − 6·22-s + 23-s + 24-s + 2·26-s + 27-s + 2·28-s + 6·29-s + 8·31-s + 32-s − 6·33-s + 36-s + 2·39-s + 10·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.235·18-s + 0.436·21-s − 1.27·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s + 0.320·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.987678651\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.987678651\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.235705960349256851451830556684, −7.985735410248379150780592398078, −7.21503654593407651256814932325, −6.22222251110775203306582284573, −5.44267515312635602226880053881, −4.71104012780114044473008537616, −4.05061488481167805028666683437, −2.81142566887325403182093960623, −2.46372583547031687479551924172, −1.09061195398127695638099915101,
1.09061195398127695638099915101, 2.46372583547031687479551924172, 2.81142566887325403182093960623, 4.05061488481167805028666683437, 4.71104012780114044473008537616, 5.44267515312635602226880053881, 6.22222251110775203306582284573, 7.21503654593407651256814932325, 7.985735410248379150780592398078, 8.235705960349256851451830556684