| L(s) = 1 | + 7·7-s − 6·11-s + 18·23-s + 25·25-s + 54·29-s − 38·37-s − 58·43-s + 49·49-s + 6·53-s + 118·67-s + 114·71-s − 42·77-s + 94·79-s + 186·107-s + 106·109-s + 222·113-s + ⋯ |
| L(s) = 1 | + 7-s − 0.545·11-s + 0.782·23-s + 25-s + 1.86·29-s − 1.02·37-s − 1.34·43-s + 49-s + 6/53·53-s + 1.76·67-s + 1.60·71-s − 0.545·77-s + 1.18·79-s + 1.73·107-s + 0.972·109-s + 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.154259717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154259717\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 + 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 18 T + p^{2} T^{2} \) |
| 29 | \( 1 - 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 118 T + p^{2} T^{2} \) |
| 71 | \( 1 - 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 - 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−9.882689182383448585989940409664, −8.688659546896135307566040057980, −8.280646871583889812073391981305, −7.26408856381416963799980929369, −6.46611811590719208224768552906, −5.15899864985601545859579675972, −4.76977705247232145076184275604, −3.39198849608765513691322632024, −2.25799251117444786074671746625, −0.948890237373402158021419236961,
0.948890237373402158021419236961, 2.25799251117444786074671746625, 3.39198849608765513691322632024, 4.76977705247232145076184275604, 5.15899864985601545859579675972, 6.46611811590719208224768552906, 7.26408856381416963799980929369, 8.280646871583889812073391981305, 8.688659546896135307566040057980, 9.882689182383448585989940409664
