| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 2·13-s − 15-s − 17-s − 4·21-s − 23-s + 25-s + 27-s + 6·29-s + 31-s + 4·35-s + 4·37-s − 2·39-s − 2·41-s + 2·43-s − 45-s + 9·47-s + 9·49-s − 51-s + 9·53-s − 6·59-s − 61-s − 4·63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 0.242·17-s − 0.872·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.179·31-s + 0.676·35-s + 0.657·37-s − 0.320·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 1.31·47-s + 9/7·49-s − 0.140·51-s + 1.23·53-s − 0.781·59-s − 0.128·61-s − 0.503·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.30465939752909, −15.72554237574642, −15.38375326183576, −14.80341821704175, −14.02118269431416, −13.66907185614693, −12.96409080894604, −12.51492882266002, −12.05350883421954, −11.36633380221514, −10.43931662579830, −10.14663897222992, −9.450710354551090, −8.983896701475094, −8.349211894375810, −7.613670495150595, −7.069967507902970, −6.482080328672738, −5.878293297397137, −4.942288334938795, −4.168720849202068, −3.618393721916394, −2.810997302640359, −2.400814799071766, −1.028429802759306, 0,
1.028429802759306, 2.400814799071766, 2.810997302640359, 3.618393721916394, 4.168720849202068, 4.942288334938795, 5.878293297397137, 6.482080328672738, 7.069967507902970, 7.613670495150595, 8.349211894375810, 8.983896701475094, 9.450710354551090, 10.14663897222992, 10.43931662579830, 11.36633380221514, 12.05350883421954, 12.51492882266002, 12.96409080894604, 13.66907185614693, 14.02118269431416, 14.80341821704175, 15.38375326183576, 15.72554237574642, 16.30465939752909
