| L(s) = 1 | + 3.22·3-s − 0.725·5-s − 7-s + 7.38·9-s + 2.21·11-s + 1.75·13-s − 2.33·15-s + 1.69·17-s + 1.80·19-s − 3.22·21-s + 8.47·23-s − 4.47·25-s + 14.1·27-s + 0.704·29-s − 6.28·31-s + 7.15·33-s + 0.725·35-s − 6.75·37-s + 5.64·39-s + 8.72·41-s − 7.33·43-s − 5.35·45-s − 10.1·47-s + 49-s + 5.45·51-s + 13.2·53-s − 1.60·55-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 0.324·5-s − 0.377·7-s + 2.46·9-s + 0.668·11-s + 0.485·13-s − 0.603·15-s + 0.410·17-s + 0.413·19-s − 0.703·21-s + 1.76·23-s − 0.894·25-s + 2.72·27-s + 0.130·29-s − 1.12·31-s + 1.24·33-s + 0.122·35-s − 1.11·37-s + 0.903·39-s + 1.36·41-s − 1.11·43-s − 0.798·45-s − 1.48·47-s + 0.142·49-s + 0.763·51-s + 1.81·53-s − 0.216·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.956195779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.956195779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 0.725T + 5T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 - 0.704T + 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 8.86T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 1.71T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.13T + 89T^{2} \) |
| 97 | \( 1 + 5.73T + 97T^{2} \) |
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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
−8.624883762763267676902330683804, −7.931609752569116952811474626039, −7.20231174541283356532482487070, −6.71685334721265055673665695736, −5.47531515480826845317353000057, −4.40693814938630610943798605596, −3.52157324204946130590892378872, −3.27592995542475511112250842648, −2.14472350964309112930358331187, −1.17514572240267795514399386605,
1.17514572240267795514399386605, 2.14472350964309112930358331187, 3.27592995542475511112250842648, 3.52157324204946130590892378872, 4.40693814938630610943798605596, 5.47531515480826845317353000057, 6.71685334721265055673665695736, 7.20231174541283356532482487070, 7.931609752569116952811474626039, 8.624883762763267676902330683804
