| L(s) = 1 | + 3-s + 0.187·5-s + 3.11·7-s + 9-s + 0.0152·11-s + 5.09·13-s + 0.187·15-s + 2.80·17-s + 4.44·19-s + 3.11·21-s + 23-s − 4.96·25-s + 27-s − 29-s − 3.85·31-s + 0.0152·33-s + 0.584·35-s − 2.44·37-s + 5.09·39-s + 4.07·41-s − 4.10·43-s + 0.187·45-s + 10.7·47-s + 2.67·49-s + 2.80·51-s + 8.69·53-s + 0.00285·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0839·5-s + 1.17·7-s + 0.333·9-s + 0.00458·11-s + 1.41·13-s + 0.0484·15-s + 0.680·17-s + 1.01·19-s + 0.678·21-s + 0.208·23-s − 0.992·25-s + 0.192·27-s − 0.185·29-s − 0.691·31-s + 0.00264·33-s + 0.0987·35-s − 0.402·37-s + 0.815·39-s + 0.636·41-s − 0.626·43-s + 0.0279·45-s + 1.57·47-s + 0.382·49-s + 0.392·51-s + 1.19·53-s + 0.000385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.707201109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.707201109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.187T + 5T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 - 0.0152T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 + 4.10T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 - 2.99T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 - 6.51T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 9.25T + 89T^{2} \) |
| 97 | \( 1 - 7.78T + 97T^{2} \) |
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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
−7.950751711441871236840343301760, −7.35026262660624117654858014416, −6.50629375023270414346000518917, −5.53160723414690057724507891770, −5.22815901785833607640177223158, −4.01991533299310400220237226339, −3.68854011926725265761331073881, −2.63548236034821687033327197991, −1.68835552744437842029053233958, −1.03110399521854255839071144618,
1.03110399521854255839071144618, 1.68835552744437842029053233958, 2.63548236034821687033327197991, 3.68854011926725265761331073881, 4.01991533299310400220237226339, 5.22815901785833607640177223158, 5.53160723414690057724507891770, 6.50629375023270414346000518917, 7.35026262660624117654858014416, 7.950751711441871236840343301760
