| L(s) = 1 | − 3·5-s + 17-s + 4·23-s + 4·25-s − 3·29-s − 8·31-s − 5·37-s − 3·41-s − 4·43-s − 8·47-s − 7·49-s + 13·53-s + 12·59-s + 15·61-s − 12·67-s + 8·71-s + 3·73-s + 4·79-s + 12·83-s − 3·85-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.242·17-s + 0.834·23-s + 4/5·25-s − 0.557·29-s − 1.43·31-s − 0.821·37-s − 0.468·41-s − 0.609·43-s − 1.16·47-s − 49-s + 1.78·53-s + 1.56·59-s + 1.92·61-s − 1.46·67-s + 0.949·71-s + 0.351·73-s + 0.450·79-s + 1.31·83-s − 0.325·85-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9527373100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9527373100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.25996281166068, −14.85480319182078, −14.67507451320275, −13.71918420931452, −13.18354233621859, −12.69555304915686, −12.07480160251666, −11.59562991182869, −11.17642732920170, −10.62226859424140, −9.942263003408594, −9.297224531394511, −8.654876741416315, −8.157966631435856, −7.658375953630202, −6.849987462557047, −6.781281577185147, −5.418765280423558, −5.278731628002622, −4.311444811315399, −3.687124259763101, −3.339976390107249, −2.353930848060974, −1.429687377401806, −0.4001509970909771,
0.4001509970909771, 1.429687377401806, 2.353930848060974, 3.339976390107249, 3.687124259763101, 4.311444811315399, 5.278731628002622, 5.418765280423558, 6.781281577185147, 6.849987462557047, 7.658375953630202, 8.157966631435856, 8.654876741416315, 9.297224531394511, 9.942263003408594, 10.62226859424140, 11.17642732920170, 11.59562991182869, 12.07480160251666, 12.69555304915686, 13.18354233621859, 13.71918420931452, 14.67507451320275, 14.85480319182078, 15.25996281166068
