| L(s) = 1 | + 3-s + 5-s + 9-s + 2·13-s + 15-s + 6·17-s − 4·19-s + 25-s + 27-s − 6·29-s + 8·31-s + 6·37-s + 2·39-s − 10·41-s − 4·43-s + 45-s + 8·47-s − 7·49-s + 6·51-s − 10·53-s − 4·57-s + 12·59-s − 6·61-s + 2·65-s + 4·67-s + 14·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.320·39-s − 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s − 0.768·61-s + 0.248·65-s + 0.488·67-s + 1.63·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.565685089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.565685089\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.24457959223627, −14.54028619215150, −14.17488460251317, −13.60940314881048, −13.11923432703615, −12.63259004255805, −12.06196657858034, −11.40149400755874, −10.84737242638187, −10.15799165226892, −9.787488648658607, −9.295240144807912, −8.449143152787199, −8.222026342657010, −7.578687255973799, −6.821694819850763, −6.284543436834485, −5.697634930247840, −5.008107003630850, −4.328070757010323, −3.545592876207188, −3.100548362527860, −2.215728270806961, −1.585646516333181, −0.7209318803185786,
0.7209318803185786, 1.585646516333181, 2.215728270806961, 3.100548362527860, 3.545592876207188, 4.328070757010323, 5.008107003630850, 5.697634930247840, 6.284543436834485, 6.821694819850763, 7.578687255973799, 8.222026342657010, 8.449143152787199, 9.295240144807912, 9.787488648658607, 10.15799165226892, 10.84737242638187, 11.40149400755874, 12.06196657858034, 12.63259004255805, 13.11923432703615, 13.60940314881048, 14.17488460251317, 14.54028619215150, 15.24457959223627
