| L(s) = 1 | − 1.45·2-s + 2.09·3-s + 0.110·4-s − 5-s − 3.04·6-s + 7-s + 2.74·8-s + 1.40·9-s + 1.45·10-s − 3.31·11-s + 0.231·12-s − 5.28·13-s − 1.45·14-s − 2.09·15-s − 4.20·16-s − 6.61·17-s − 2.03·18-s + 1.66·19-s − 0.110·20-s + 2.09·21-s + 4.82·22-s + 23-s + 5.75·24-s + 25-s + 7.67·26-s − 3.35·27-s + 0.110·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 1.21·3-s + 0.0552·4-s − 0.447·5-s − 1.24·6-s + 0.377·7-s + 0.970·8-s + 0.467·9-s + 0.459·10-s − 1.00·11-s + 0.0669·12-s − 1.46·13-s − 0.388·14-s − 0.541·15-s − 1.05·16-s − 1.60·17-s − 0.480·18-s + 0.381·19-s − 0.0247·20-s + 0.457·21-s + 1.02·22-s + 0.208·23-s + 1.17·24-s + 0.200·25-s + 1.50·26-s − 0.644·27-s + 0.0208·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 11 | \( 1 + 3.31T + 11T^{2} \) |
| 13 | \( 1 + 5.28T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 + 8.26T + 41T^{2} \) |
| 43 | \( 1 - 7.03T + 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 + 9.69T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 4.79T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 0.525T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 8.27T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.537489404696774004713991421352, −8.958801851182830455114509498644, −8.179408040295034094594482193587, −7.68915655322038751432335352796, −6.93035522994283372243144969155, −5.08932366892090610162524984249, −4.35415950460876924232025728972, −2.91232825411454401932598666461, −2.00241919486724416190190269685, 0,
2.00241919486724416190190269685, 2.91232825411454401932598666461, 4.35415950460876924232025728972, 5.08932366892090610162524984249, 6.93035522994283372243144969155, 7.68915655322038751432335352796, 8.179408040295034094594482193587, 8.958801851182830455114509498644, 9.537489404696774004713991421352
