| L(s) = 1 | − 9·3-s + 6·5-s + 49·7-s + 81·9-s − 108·11-s − 346·13-s − 54·15-s − 1.39e3·17-s − 1.01e3·19-s − 441·21-s − 1.53e3·23-s − 3.08e3·25-s − 729·27-s − 3.76e3·29-s − 736·31-s + 972·33-s + 294·35-s + 2.05e3·37-s + 3.11e3·39-s − 1.55e4·41-s + 1.10e4·43-s + 486·45-s + 4.56e3·47-s + 2.40e3·49-s + 1.25e4·51-s − 7.96e3·53-s − 648·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.107·5-s + 0.377·7-s + 1/3·9-s − 0.269·11-s − 0.567·13-s − 0.0619·15-s − 1.17·17-s − 0.643·19-s − 0.218·21-s − 0.605·23-s − 0.988·25-s − 0.192·27-s − 0.830·29-s − 0.137·31-s + 0.155·33-s + 0.0405·35-s + 0.246·37-s + 0.327·39-s − 1.44·41-s + 0.910·43-s + 0.0357·45-s + 0.301·47-s + 1/7·49-s + 0.677·51-s − 0.389·53-s − 0.0288·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 11 | \( 1 + 108 T + p^{5} T^{2} \) |
| 13 | \( 1 + 346 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1398 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1012 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1536 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3762 T + p^{5} T^{2} \) |
| 31 | \( 1 + 736 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2054 T + p^{5} T^{2} \) |
| 41 | \( 1 + 15534 T + p^{5} T^{2} \) |
| 43 | \( 1 - 11036 T + p^{5} T^{2} \) |
| 47 | \( 1 - 4560 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7962 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7020 T + p^{5} T^{2} \) |
| 61 | \( 1 - 26870 T + p^{5} T^{2} \) |
| 67 | \( 1 - 52148 T + p^{5} T^{2} \) |
| 71 | \( 1 + 2544 T + p^{5} T^{2} \) |
| 73 | \( 1 + 9766 T + p^{5} T^{2} \) |
| 79 | \( 1 - 68672 T + p^{5} T^{2} \) |
| 83 | \( 1 + 61668 T + p^{5} T^{2} \) |
| 89 | \( 1 + 41454 T + p^{5} T^{2} \) |
| 97 | \( 1 + 111262 T + p^{5} T^{2} \) |
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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
−12.70987724973401599935095432225, −11.61040791791882310951281541863, −10.69033219718494232263830482518, −9.521831963978602286033636957168, −8.124799006866335743308090813313, −6.82080511089064574446156943468, −5.51957437080052593999409368883, −4.22206514312905283403621173119, −2.07201297117303168597525594645, 0,
2.07201297117303168597525594645, 4.22206514312905283403621173119, 5.51957437080052593999409368883, 6.82080511089064574446156943468, 8.124799006866335743308090813313, 9.521831963978602286033636957168, 10.69033219718494232263830482518, 11.61040791791882310951281541863, 12.70987724973401599935095432225
