| L(s) = 1 | − 1.82·2-s − 4.65·4-s + 7.31·5-s − 11.1·7-s + 23.1·8-s − 13.3·10-s + 16.2·11-s − 13.0·13-s + 20.4·14-s − 5.05·16-s − 33.8·17-s − 6.11·19-s − 34.0·20-s − 29.6·22-s − 23·23-s − 71.5·25-s + 23.8·26-s + 52.0·28-s − 84.6·29-s − 236.·31-s − 175.·32-s + 61.9·34-s − 81.7·35-s − 63.6·37-s + 11.1·38-s + 169.·40-s − 75.5·41-s + ⋯ |
| L(s) = 1 | − 0.646·2-s − 0.582·4-s + 0.654·5-s − 0.603·7-s + 1.02·8-s − 0.422·10-s + 0.444·11-s − 0.277·13-s + 0.389·14-s − 0.0790·16-s − 0.483·17-s − 0.0738·19-s − 0.380·20-s − 0.287·22-s − 0.208·23-s − 0.572·25-s + 0.179·26-s + 0.351·28-s − 0.542·29-s − 1.37·31-s − 0.971·32-s + 0.312·34-s − 0.394·35-s − 0.282·37-s + 0.0477·38-s + 0.669·40-s − 0.287·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 1.82T + 8T^{2} \) |
| 5 | \( 1 - 7.31T + 125T^{2} \) |
| 7 | \( 1 + 11.1T + 343T^{2} \) |
| 11 | \( 1 - 16.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.11T + 6.85e3T^{2} \) |
| 29 | \( 1 + 84.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 63.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 75.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 260.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 44.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 466.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 906.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 920.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−11.25678540656775151989507403257, −10.12891452228298300512903778709, −9.501259197425938196948591801396, −8.723152197346698921612891831737, −7.50544644875786313813837304064, −6.32364180697561043563539853805, −5.07612916495418896418534243292, −3.69594002042427584732786389401, −1.80474675132387779577310287004, 0,
1.80474675132387779577310287004, 3.69594002042427584732786389401, 5.07612916495418896418534243292, 6.32364180697561043563539853805, 7.50544644875786313813837304064, 8.723152197346698921612891831737, 9.501259197425938196948591801396, 10.12891452228298300512903778709, 11.25678540656775151989507403257
