| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 4·11-s + 6·13-s + 14-s + 16-s − 2·17-s + 4·19-s − 20-s + 4·22-s + 23-s + 25-s − 6·26-s − 28-s + 2·29-s − 8·31-s − 32-s + 2·34-s + 35-s − 2·37-s − 4·38-s + 40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.23388790478504, −15.89299104776924, −15.59354060290170, −14.88425200936766, −14.18242496980943, −13.28587914085467, −13.23369206970487, −12.40788745204283, −11.73922115293012, −11.09303623256618, −10.76535412887505, −10.18728995431144, −9.441463930797795, −8.801821405758043, −8.438785510031590, −7.610403294808575, −7.299961698759296, −6.435038372901019, −5.834683474918776, −5.199715538191292, −4.259702716954991, −3.424171544144559, −2.947532778188614, −1.927978425752178, −0.9948967426958785, 0,
0.9948967426958785, 1.927978425752178, 2.947532778188614, 3.424171544144559, 4.259702716954991, 5.199715538191292, 5.834683474918776, 6.435038372901019, 7.299961698759296, 7.610403294808575, 8.438785510031590, 8.801821405758043, 9.441463930797795, 10.18728995431144, 10.76535412887505, 11.09303623256618, 11.73922115293012, 12.40788745204283, 13.23369206970487, 13.28587914085467, 14.18242496980943, 14.88425200936766, 15.59354060290170, 15.89299104776924, 16.23388790478504
