| L(s) = 1 | − 3-s − 2.54·5-s + 0.525·7-s + 9-s + 5.59·11-s + 1.57·13-s + 2.54·15-s + 4.20·17-s + 3.78·19-s − 0.525·21-s − 2.45·23-s + 1.47·25-s − 27-s − 4.24·29-s − 5.76·31-s − 5.59·33-s − 1.33·35-s − 2.67·37-s − 1.57·39-s + 10.7·41-s + 8.36·43-s − 2.54·45-s + 4.77·47-s − 6.72·49-s − 4.20·51-s − 2.34·53-s − 14.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·5-s + 0.198·7-s + 0.333·9-s + 1.68·11-s + 0.437·13-s + 0.656·15-s + 1.02·17-s + 0.869·19-s − 0.114·21-s − 0.511·23-s + 0.294·25-s − 0.192·27-s − 0.787·29-s − 1.03·31-s − 0.973·33-s − 0.225·35-s − 0.439·37-s − 0.252·39-s + 1.68·41-s + 1.27·43-s − 0.379·45-s + 0.697·47-s − 0.960·49-s − 0.589·51-s − 0.322·53-s − 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429644770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429644770\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.54T + 5T^{2} \) |
| 7 | \( 1 - 0.525T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 8.36T + 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 + 2.34T + 53T^{2} \) |
| 59 | \( 1 + 5.25T + 59T^{2} \) |
| 61 | \( 1 + 5.02T + 61T^{2} \) |
| 67 | \( 1 - 8.91T + 67T^{2} \) |
| 71 | \( 1 + 5.10T + 71T^{2} \) |
| 73 | \( 1 + 2.16T + 73T^{2} \) |
| 79 | \( 1 + 2.52T + 79T^{2} \) |
| 83 | \( 1 + 4.01T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.353627035601448541664391026083, −7.52349742813703569036572205720, −7.20638312393963320040397033487, −6.11111599064667741010451963980, −5.63437765822439199114592049363, −4.49326936961276928360565776315, −3.88924517596389503933242684161, −3.30836339216343892727685892117, −1.67392798564421509239599682291, −0.75241076168333314313769055560,
0.75241076168333314313769055560, 1.67392798564421509239599682291, 3.30836339216343892727685892117, 3.88924517596389503933242684161, 4.49326936961276928360565776315, 5.63437765822439199114592049363, 6.11111599064667741010451963980, 7.20638312393963320040397033487, 7.52349742813703569036572205720, 8.353627035601448541664391026083
