| L(s) = 1 | − 3-s − 5-s + 3.86·7-s + 9-s + 5.64·11-s + 1.78·13-s + 15-s − 2.64·17-s + 5.39·19-s − 3.86·21-s + 4.97·23-s + 25-s − 27-s − 3.10·29-s + 9.72·31-s − 5.64·33-s − 3.86·35-s + 0.754·37-s − 1.78·39-s − 2.64·41-s + 2.89·43-s − 45-s + 0.144·47-s + 7.92·49-s + 2.64·51-s − 6.68·53-s − 5.64·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.46·7-s + 0.333·9-s + 1.70·11-s + 0.494·13-s + 0.258·15-s − 0.642·17-s + 1.23·19-s − 0.843·21-s + 1.03·23-s + 0.200·25-s − 0.192·27-s − 0.577·29-s + 1.74·31-s − 0.982·33-s − 0.653·35-s + 0.124·37-s − 0.285·39-s − 0.412·41-s + 0.442·43-s − 0.149·45-s + 0.0210·47-s + 1.13·49-s + 0.370·51-s − 0.917·53-s − 0.761·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.190402647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.190402647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 - 9.72T + 31T^{2} \) |
| 37 | \( 1 - 0.754T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 0.144T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 - 6.29T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 71 | \( 1 + 6.71T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 3.85T + 83T^{2} \) |
| 89 | \( 1 - 4.78T + 89T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−8.501471604714844388237429280270, −7.64502456625472839090881343226, −6.99309598706464707242247017703, −6.27857413295736279777057565007, −5.39930252973703395806277893078, −4.58518766389745661953675564560, −4.12862281487638903233178295732, −3.04783235279062203807698693167, −1.58778804048236144925447393457, −1.00640211261018419478951482376,
1.00640211261018419478951482376, 1.58778804048236144925447393457, 3.04783235279062203807698693167, 4.12862281487638903233178295732, 4.58518766389745661953675564560, 5.39930252973703395806277893078, 6.27857413295736279777057565007, 6.99309598706464707242247017703, 7.64502456625472839090881343226, 8.501471604714844388237429280270
