| L(s) = 1 | + 0.150·3-s + 2.87·5-s + 2.68·7-s − 2.97·9-s − 0.368·11-s − 2.43·13-s + 0.432·15-s − 3.34·17-s − 0.377·19-s + 0.403·21-s + 0.915·23-s + 3.28·25-s − 0.897·27-s − 2.06·29-s − 5.88·31-s − 0.0552·33-s + 7.72·35-s − 7.95·37-s − 0.366·39-s − 1.24·41-s − 5.58·43-s − 8.57·45-s + 12.4·47-s + 0.207·49-s − 0.502·51-s − 11.7·53-s − 1.05·55-s + ⋯ |
| L(s) = 1 | + 0.0866·3-s + 1.28·5-s + 1.01·7-s − 0.992·9-s − 0.110·11-s − 0.676·13-s + 0.111·15-s − 0.811·17-s − 0.0866·19-s + 0.0879·21-s + 0.190·23-s + 0.657·25-s − 0.172·27-s − 0.384·29-s − 1.05·31-s − 0.00961·33-s + 1.30·35-s − 1.30·37-s − 0.0586·39-s − 0.193·41-s − 0.851·43-s − 1.27·45-s + 1.81·47-s + 0.0296·49-s − 0.0703·51-s − 1.62·53-s − 0.142·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 0.150T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 + 0.368T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 + 0.377T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 + 2.06T + 29T^{2} \) |
| 31 | \( 1 + 5.88T + 31T^{2} \) |
| 37 | \( 1 + 7.95T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 5.58T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 0.641T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.38111002981726631671764717295, −6.72873512306226283915255678628, −5.88333946753795683223991577148, −5.34448514250356064716574677373, −4.89342034789023537148270506091, −3.88083088379657663167097738270, −2.81197915922713338486402334851, −2.14565244107419497358240562089, −1.54143791720483931818026048135, 0,
1.54143791720483931818026048135, 2.14565244107419497358240562089, 2.81197915922713338486402334851, 3.88083088379657663167097738270, 4.89342034789023537148270506091, 5.34448514250356064716574677373, 5.88333946753795683223991577148, 6.72873512306226283915255678628, 7.38111002981726631671764717295
