| L(s) = 1 | − 2.90·3-s − 1.66·5-s − 2.42·7-s + 5.46·9-s − 0.493·11-s − 13-s + 4.84·15-s − 1.59·17-s − 4.35·19-s + 7.05·21-s − 5.41·23-s − 2.22·25-s − 7.18·27-s + 29-s − 0.312·31-s + 1.43·33-s + 4.03·35-s + 9.46·37-s + 2.90·39-s − 10.7·41-s − 1.06·43-s − 9.10·45-s − 11.4·47-s − 1.11·49-s + 4.64·51-s + 12.7·53-s + 0.822·55-s + ⋯ |
| L(s) = 1 | − 1.68·3-s − 0.744·5-s − 0.916·7-s + 1.82·9-s − 0.148·11-s − 0.277·13-s + 1.25·15-s − 0.386·17-s − 0.998·19-s + 1.54·21-s − 1.12·23-s − 0.445·25-s − 1.38·27-s + 0.185·29-s − 0.0560·31-s + 0.250·33-s + 0.682·35-s + 1.55·37-s + 0.465·39-s − 1.68·41-s − 0.162·43-s − 1.35·45-s − 1.67·47-s − 0.159·49-s + 0.650·51-s + 1.75·53-s + 0.110·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01490925618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01490925618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 + 0.493T + 11T^{2} \) |
| 17 | \( 1 + 1.59T + 17T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 31 | \( 1 + 0.312T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 - 3.92T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 9.87T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−7.923637204663548894497332392872, −7.17143841889283015763810482268, −6.43921187244029641624334549670, −6.12234516120449327308880602582, −5.28198700006943151670882690189, −4.43143115809042830741816404729, −3.97662790112058642921524657200, −2.85343993912719351994385581073, −1.57952339424998579736439862456, −0.06941363171385168480022107034,
0.06941363171385168480022107034, 1.57952339424998579736439862456, 2.85343993912719351994385581073, 3.97662790112058642921524657200, 4.43143115809042830741816404729, 5.28198700006943151670882690189, 6.12234516120449327308880602582, 6.43921187244029641624334549670, 7.17143841889283015763810482268, 7.923637204663548894497332392872
