| L(s) = 1 | + 2-s + 1.19·3-s + 4-s − 0.354·5-s + 1.19·6-s + 1.00·7-s + 8-s − 1.58·9-s − 0.354·10-s + 5.05·11-s + 1.19·12-s − 0.717·13-s + 1.00·14-s − 0.422·15-s + 16-s − 3.01·17-s − 1.58·18-s + 0.552·19-s − 0.354·20-s + 1.19·21-s + 5.05·22-s + 3.54·23-s + 1.19·24-s − 4.87·25-s − 0.717·26-s − 5.45·27-s + 1.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.687·3-s + 0.5·4-s − 0.158·5-s + 0.485·6-s + 0.380·7-s + 0.353·8-s − 0.527·9-s − 0.112·10-s + 1.52·11-s + 0.343·12-s − 0.199·13-s + 0.268·14-s − 0.109·15-s + 0.250·16-s − 0.731·17-s − 0.373·18-s + 0.126·19-s − 0.0793·20-s + 0.261·21-s + 1.07·22-s + 0.738·23-s + 0.242·24-s − 0.974·25-s − 0.140·26-s − 1.04·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.190700099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.190700099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 0.354T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 + 0.717T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 - 0.552T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 1.06T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 - 0.0104T + 59T^{2} \) |
| 61 | \( 1 - 5.71T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 - 0.163T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 + 3.85T + 89T^{2} \) |
| 97 | \( 1 - 9.39T + 97T^{2} \) |
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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
−8.432676448498318644367792698708, −7.73916923995770058940004574891, −6.86521509214044882326235957417, −6.27229819113825781789889367203, −5.43482832859313731959421981221, −4.40848485693646661433026199187, −3.97587989495692150826830313704, −2.96156363967033263423476283746, −2.27330192394141188245009309257, −1.08041598455573559543794940153,
1.08041598455573559543794940153, 2.27330192394141188245009309257, 2.96156363967033263423476283746, 3.97587989495692150826830313704, 4.40848485693646661433026199187, 5.43482832859313731959421981221, 6.27229819113825781789889367203, 6.86521509214044882326235957417, 7.73916923995770058940004574891, 8.432676448498318644367792698708
