| L(s) = 1 | − 3-s + 3.42·5-s − 3.44·7-s + 9-s + 3.41·11-s − 2.00·13-s − 3.42·15-s − 3.07·17-s + 0.565·19-s + 3.44·21-s − 7.40·23-s + 6.76·25-s − 27-s + 9.73·29-s + 2.62·31-s − 3.41·33-s − 11.8·35-s − 0.527·37-s + 2.00·39-s + 3.19·41-s − 7.98·43-s + 3.42·45-s + 6.96·47-s + 4.84·49-s + 3.07·51-s + 2.40·53-s + 11.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.53·5-s − 1.30·7-s + 0.333·9-s + 1.02·11-s − 0.556·13-s − 0.885·15-s − 0.746·17-s + 0.129·19-s + 0.751·21-s − 1.54·23-s + 1.35·25-s − 0.192·27-s + 1.80·29-s + 0.471·31-s − 0.593·33-s − 1.99·35-s − 0.0866·37-s + 0.321·39-s + 0.499·41-s − 1.21·43-s + 0.511·45-s + 1.01·47-s + 0.692·49-s + 0.430·51-s + 0.330·53-s + 1.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798118492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798118492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 2.00T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 - 0.565T + 19T^{2} \) |
| 23 | \( 1 + 7.40T + 23T^{2} \) |
| 29 | \( 1 - 9.73T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 + 7.98T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 + 6.23T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 0.623T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 - 4.25T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.71463282298998300961872071722, −6.67285767269346485954178256443, −6.37811935119215727008963146736, −6.08136954442910256911510589695, −5.12318188689092817603672474054, −4.41645466831236084010772720096, −3.47285908288475013179070647817, −2.53454766841509965318641306742, −1.81704421664154926594113129424, −0.67674519476775632653716304034,
0.67674519476775632653716304034, 1.81704421664154926594113129424, 2.53454766841509965318641306742, 3.47285908288475013179070647817, 4.41645466831236084010772720096, 5.12318188689092817603672474054, 6.08136954442910256911510589695, 6.37811935119215727008963146736, 6.67285767269346485954178256443, 7.71463282298998300961872071722
