| L(s) = 1 | + 2.40·3-s + 0.936·5-s − 3.55·7-s + 2.80·9-s − 1.35·11-s − 2.20·13-s + 2.25·15-s + 1.19·17-s + 19-s − 8.56·21-s − 5.26·23-s − 4.12·25-s − 0.478·27-s + 9.37·29-s + 0.532·31-s − 3.25·33-s − 3.33·35-s + 4.51·37-s − 5.30·39-s − 8.05·41-s + 12.1·43-s + 2.62·45-s − 0.711·47-s + 5.64·49-s + 2.87·51-s − 12.0·53-s − 1.26·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 0.418·5-s − 1.34·7-s + 0.933·9-s − 0.407·11-s − 0.611·13-s + 0.582·15-s + 0.289·17-s + 0.229·19-s − 1.86·21-s − 1.09·23-s − 0.824·25-s − 0.0921·27-s + 1.74·29-s + 0.0956·31-s − 0.566·33-s − 0.562·35-s + 0.742·37-s − 0.850·39-s − 1.25·41-s + 1.84·43-s + 0.391·45-s − 0.103·47-s + 0.805·49-s + 0.402·51-s − 1.65·53-s − 0.170·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 0.936T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 23 | \( 1 + 5.26T + 23T^{2} \) |
| 29 | \( 1 - 9.37T + 29T^{2} \) |
| 31 | \( 1 - 0.532T + 31T^{2} \) |
| 37 | \( 1 - 4.51T + 37T^{2} \) |
| 41 | \( 1 + 8.05T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 0.711T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 7.85T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.91T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 7.79T + 73T^{2} \) |
| 83 | \( 1 + 8.02T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.87571427765272905439221679884, −7.16614947321715031026725427060, −6.29684971409721833457288076301, −5.78365590840919317549184436920, −4.63485142112025014695930450713, −3.84818828435509801297282838038, −2.90758470076528344980299201257, −2.70987242273639945767201789424, −1.60900310338533282701315999410, 0,
1.60900310338533282701315999410, 2.70987242273639945767201789424, 2.90758470076528344980299201257, 3.84818828435509801297282838038, 4.63485142112025014695930450713, 5.78365590840919317549184436920, 6.29684971409721833457288076301, 7.16614947321715031026725427060, 7.87571427765272905439221679884
