L(s) = 1 | − 2·2-s + 4·4-s − 10.4·5-s − 8·8-s + 20.8·10-s − 61.1·11-s + 59.2·13-s + 16·16-s + 20.4·17-s + 80.3·19-s − 41.7·20-s + 122.·22-s + 158.·23-s − 15.8·25-s − 118.·26-s + 85.1·29-s − 243.·31-s − 32·32-s − 40.9·34-s + 290.·37-s − 160.·38-s + 83.5·40-s − 168·41-s + 7.62·43-s − 244.·44-s − 316.·46-s − 169.·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.934·5-s − 0.353·8-s + 0.660·10-s − 1.67·11-s + 1.26·13-s + 0.250·16-s + 0.291·17-s + 0.970·19-s − 0.467·20-s + 1.18·22-s + 1.43·23-s − 0.127·25-s − 0.893·26-s + 0.545·29-s − 1.41·31-s − 0.176·32-s − 0.206·34-s + 1.29·37-s − 0.685·38-s + 0.330·40-s − 0.639·41-s + 0.0270·43-s − 0.837·44-s − 1.01·46-s − 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 10.4T + 125T^{2} \) |
| 11 | \( 1 + 61.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.62T + 7.95e4T^{2} \) |
| 47 | \( 1 + 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 805.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 33.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 768.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−9.253090624008808895699411053299, −8.350336307876207459631768824872, −7.77240553284491167487940665479, −7.09957180251424029552736089372, −5.86610041828587300779013465969, −4.96672594782918422361376718589, −3.61382417421161474660633470060, −2.78010997241749435217309452774, −1.19454548994668685365480765120, 0,
1.19454548994668685365480765120, 2.78010997241749435217309452774, 3.61382417421161474660633470060, 4.96672594782918422361376718589, 5.86610041828587300779013465969, 7.09957180251424029552736089372, 7.77240553284491167487940665479, 8.350336307876207459631768824872, 9.253090624008808895699411053299