| L(s) = 1 | + 1.21·2-s − 0.522·4-s − 1.83·5-s − 2.18·7-s − 3.06·8-s − 2.22·10-s + 4.11·11-s + 2.89·13-s − 2.65·14-s − 2.68·16-s + 2.75·17-s + 4.55·19-s + 0.955·20-s + 4.99·22-s + 8.48·23-s − 1.64·25-s + 3.52·26-s + 1.14·28-s + 6.07·29-s − 9.34·31-s + 2.87·32-s + 3.35·34-s + 4·35-s + 4.54·37-s + 5.53·38-s + 5.61·40-s + 8.30·41-s + ⋯ |
| L(s) = 1 | + 0.859·2-s − 0.261·4-s − 0.818·5-s − 0.825·7-s − 1.08·8-s − 0.703·10-s + 1.23·11-s + 0.803·13-s − 0.709·14-s − 0.670·16-s + 0.668·17-s + 1.04·19-s + 0.213·20-s + 1.06·22-s + 1.76·23-s − 0.329·25-s + 0.691·26-s + 0.215·28-s + 1.12·29-s − 1.67·31-s + 0.507·32-s + 0.574·34-s + 0.676·35-s + 0.747·37-s + 0.898·38-s + 0.887·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815921245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815921245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 4.11T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 - 6.07T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 - 4.21T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 - 5.62T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 + 5.92T + 73T^{2} \) |
| 79 | \( 1 - 3.59T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 1.27T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.423621605086002091996400973041, −9.263588288445125098719545001979, −8.171337821607030095391982964211, −7.13855808755334152283703814515, −6.32601891180331723064863203969, −5.49966154695864397403655859100, −4.44537529206607728372071914071, −3.58771695569346908614477976712, −3.14011642276793315234147660860, −0.934638777988258646526324652742,
0.934638777988258646526324652742, 3.14011642276793315234147660860, 3.58771695569346908614477976712, 4.44537529206607728372071914071, 5.49966154695864397403655859100, 6.32601891180331723064863203969, 7.13855808755334152283703814515, 8.171337821607030095391982964211, 9.263588288445125098719545001979, 9.423621605086002091996400973041
