L(s) = 1 | − 2.30·2-s − 3-s + 3.29·4-s − 0.193·5-s + 2.30·6-s + 1.10·7-s − 2.97·8-s + 9-s + 0.446·10-s + 0.301·11-s − 3.29·12-s − 5.80·13-s − 2.53·14-s + 0.193·15-s + 0.256·16-s + 17-s − 2.30·18-s + 4.49·19-s − 0.638·20-s − 1.10·21-s − 0.692·22-s + 7.40·23-s + 2.97·24-s − 4.96·25-s + 13.3·26-s − 27-s + 3.63·28-s + ⋯ |
L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.64·4-s − 0.0867·5-s + 0.939·6-s + 0.417·7-s − 1.05·8-s + 0.333·9-s + 0.141·10-s + 0.0907·11-s − 0.950·12-s − 1.60·13-s − 0.678·14-s + 0.0500·15-s + 0.0641·16-s + 0.242·17-s − 0.542·18-s + 1.03·19-s − 0.142·20-s − 0.240·21-s − 0.147·22-s + 1.54·23-s + 0.607·24-s − 0.992·25-s + 2.61·26-s − 0.192·27-s + 0.686·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 5 | \( 1 + 0.193T + 5T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 - 0.301T + 11T^{2} \) |
| 13 | \( 1 + 5.80T + 13T^{2} \) |
| 19 | \( 1 - 4.49T + 19T^{2} \) |
| 23 | \( 1 - 7.40T + 23T^{2} \) |
| 29 | \( 1 + 7.01T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 - 5.84T + 43T^{2} \) |
| 47 | \( 1 - 9.29T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.447T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 5.44T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−7.941588939194407492121689555651, −7.39571892949594851327627973494, −7.14669620796976021374797831930, −6.00189717891524724416034619736, −5.19403770186582899163939840011, −4.43478788009998410275341544199, −3.06507185180467882369908046721, −2.05676467790487739530696697804, −1.10525189060539783273484370519, 0,
1.10525189060539783273484370519, 2.05676467790487739530696697804, 3.06507185180467882369908046721, 4.43478788009998410275341544199, 5.19403770186582899163939840011, 6.00189717891524724416034619736, 7.14669620796976021374797831930, 7.39571892949594851327627973494, 7.941588939194407492121689555651