| L(s) = 1 | − 1.57·2-s − 2.20·3-s + 0.479·4-s + 3.05·5-s + 3.46·6-s − 0.157·7-s + 2.39·8-s + 1.84·9-s − 4.80·10-s + 4.18·11-s − 1.05·12-s − 4.86·13-s + 0.248·14-s − 6.71·15-s − 4.72·16-s − 2.33·17-s − 2.91·18-s + 3.05·19-s + 1.46·20-s + 0.347·21-s − 6.59·22-s + 4.14·23-s − 5.27·24-s + 4.31·25-s + 7.65·26-s + 2.53·27-s − 0.0757·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 1.27·3-s + 0.239·4-s + 1.36·5-s + 1.41·6-s − 0.0596·7-s + 0.846·8-s + 0.616·9-s − 1.51·10-s + 1.26·11-s − 0.304·12-s − 1.34·13-s + 0.0664·14-s − 1.73·15-s − 1.18·16-s − 0.567·17-s − 0.686·18-s + 0.700·19-s + 0.327·20-s + 0.0758·21-s − 1.40·22-s + 0.865·23-s − 1.07·24-s + 0.862·25-s + 1.50·26-s + 0.487·27-s − 0.0143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6457965698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6457965698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 + 0.157T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + 2.33T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 + 0.0693T + 41T^{2} \) |
| 43 | \( 1 + 6.65T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 0.129T + 53T^{2} \) |
| 59 | \( 1 + 2.75T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 - 8.84T + 67T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 - 1.08T + 83T^{2} \) |
| 89 | \( 1 - 6.86T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.73399694498109416887423514372, −7.03538046768345152146135101677, −6.55721769715412117376675576471, −5.85767673653977679390882323119, −5.03443140693491702881638970606, −4.76317502227296300290639362186, −3.46610379589729272941380362864, −2.12959905323806901241003848382, −1.52878448650152188012121957779, −0.52939983119091007275480057143,
0.52939983119091007275480057143, 1.52878448650152188012121957779, 2.12959905323806901241003848382, 3.46610379589729272941380362864, 4.76317502227296300290639362186, 5.03443140693491702881638970606, 5.85767673653977679390882323119, 6.55721769715412117376675576471, 7.03538046768345152146135101677, 7.73399694498109416887423514372
