L(s) = 1 | − 2-s − 3.27·3-s + 4-s + 2.11·5-s + 3.27·6-s − 0.124·7-s − 8-s + 7.75·9-s − 2.11·10-s − 5.51·11-s − 3.27·12-s + 4.44·13-s + 0.124·14-s − 6.95·15-s + 16-s − 0.642·17-s − 7.75·18-s + 0.147·19-s + 2.11·20-s + 0.409·21-s + 5.51·22-s + 3.37·23-s + 3.27·24-s − 0.508·25-s − 4.44·26-s − 15.6·27-s − 0.124·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.89·3-s + 0.5·4-s + 0.947·5-s + 1.33·6-s − 0.0472·7-s − 0.353·8-s + 2.58·9-s − 0.670·10-s − 1.66·11-s − 0.946·12-s + 1.23·13-s + 0.0333·14-s − 1.79·15-s + 0.250·16-s − 0.155·17-s − 1.82·18-s + 0.0338·19-s + 0.473·20-s + 0.0894·21-s + 1.17·22-s + 0.704·23-s + 0.669·24-s − 0.101·25-s − 0.871·26-s − 3.00·27-s − 0.0236·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 2.11T + 5T^{2} \) |
| 7 | \( 1 + 0.124T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 + 0.642T + 17T^{2} \) |
| 19 | \( 1 - 0.147T + 19T^{2} \) |
| 23 | \( 1 - 3.37T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 + 7.36T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 9.35T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 5.37T + 97T^{2} \) |
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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
−8.076180898826760411747338792397, −7.10672293233338863391879230740, −6.50616621284085703260976474846, −5.91780738164044301065638257858, −5.28432555120921881498894053682, −4.73464161865317399472050683863, −3.32359467798954020857131332253, −2.02950792366550579517653151253, −1.13532261385657021658854009011, 0,
1.13532261385657021658854009011, 2.02950792366550579517653151253, 3.32359467798954020857131332253, 4.73464161865317399472050683863, 5.28432555120921881498894053682, 5.91780738164044301065638257858, 6.50616621284085703260976474846, 7.10672293233338863391879230740, 8.076180898826760411747338792397