| L(s) = 1 | + 2.83·5-s − 17.1·7-s + 48.7·11-s − 64.6·13-s + 118.·17-s − 99.4·19-s + 57.9·23-s − 116.·25-s + 27.3·29-s + 58.0·31-s − 48.5·35-s + 133.·37-s − 195.·41-s + 351.·43-s − 568.·47-s − 50.5·49-s + 208.·53-s + 138.·55-s + 810.·59-s − 583.·61-s − 183.·65-s − 22.5·67-s − 218.·71-s + 273.·73-s − 833.·77-s + 830.·79-s + 650.·83-s + ⋯ |
| L(s) = 1 | + 0.253·5-s − 0.923·7-s + 1.33·11-s − 1.37·13-s + 1.69·17-s − 1.20·19-s + 0.525·23-s − 0.935·25-s + 0.175·29-s + 0.336·31-s − 0.234·35-s + 0.592·37-s − 0.743·41-s + 1.24·43-s − 1.76·47-s − 0.147·49-s + 0.539·53-s + 0.338·55-s + 1.78·59-s − 1.22·61-s − 0.349·65-s − 0.0411·67-s − 0.365·71-s + 0.438·73-s − 1.23·77-s + 1.18·79-s + 0.860·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.860901833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.860901833\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.83T + 125T^{2} \) |
| 7 | \( 1 + 17.1T + 343T^{2} \) |
| 11 | \( 1 - 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 27.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 208.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 810.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 22.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 218.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 273.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 830.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 650.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 633.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 290.T + 9.12e5T^{2} \) |
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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
−9.154365779240825450573238640884, −8.163395360666096049974819343807, −7.27923673416574191172243618256, −6.52465330752023336358341945922, −5.87884136957286228115887027416, −4.84066055568073110158495002507, −3.85820349411845819971169545109, −3.01376852115489898806701135658, −1.90108200187886890342832250900, −0.64185484872344849017196333958,
0.64185484872344849017196333958, 1.90108200187886890342832250900, 3.01376852115489898806701135658, 3.85820349411845819971169545109, 4.84066055568073110158495002507, 5.87884136957286228115887027416, 6.52465330752023336358341945922, 7.27923673416574191172243618256, 8.163395360666096049974819343807, 9.154365779240825450573238640884
