| L(s) = 1 | + 3.25·3-s + 1.44·5-s + 2.24·7-s + 7.58·9-s − 4.76·11-s − 3.04·13-s + 4.69·15-s + 3.04·17-s + 19-s + 7.31·21-s + 8.62·23-s − 2.92·25-s + 14.9·27-s + 0.0441·29-s + 8.22·31-s − 15.5·33-s + 3.23·35-s − 6.87·37-s − 9.89·39-s + 7.26·41-s + 5.96·43-s + 10.9·45-s + 9.10·47-s − 1.95·49-s + 9.91·51-s − 13.3·53-s − 6.87·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 0.644·5-s + 0.849·7-s + 2.52·9-s − 1.43·11-s − 0.843·13-s + 1.21·15-s + 0.738·17-s + 0.229·19-s + 1.59·21-s + 1.79·23-s − 0.584·25-s + 2.87·27-s + 0.00820·29-s + 1.47·31-s − 2.70·33-s + 0.547·35-s − 1.13·37-s − 1.58·39-s + 1.13·41-s + 0.909·43-s + 1.63·45-s + 1.32·47-s − 0.278·49-s + 1.38·51-s − 1.84·53-s − 0.926·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.211162436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.211162436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 23 | \( 1 - 8.62T + 23T^{2} \) |
| 29 | \( 1 - 0.0441T + 29T^{2} \) |
| 31 | \( 1 - 8.22T + 31T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 - 7.26T + 41T^{2} \) |
| 43 | \( 1 - 5.96T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 2.28T + 71T^{2} \) |
| 73 | \( 1 - 0.946T + 73T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.968953589769764507280427576184, −7.64856251348586220935362236915, −7.08635735202684500100811929205, −5.87255004940594541375251280128, −4.92421953333452927631638141607, −4.55334010491846491934491682129, −3.27079735348287567836692749260, −2.75151997477407487587139141788, −2.12060624621872051528209472605, −1.19508065267333638113124350578,
1.19508065267333638113124350578, 2.12060624621872051528209472605, 2.75151997477407487587139141788, 3.27079735348287567836692749260, 4.55334010491846491934491682129, 4.92421953333452927631638141607, 5.87255004940594541375251280128, 7.08635735202684500100811929205, 7.64856251348586220935362236915, 7.968953589769764507280427576184
