| L(s) = 1 | − 18·3-s + 25·5-s − 242·7-s + 81·9-s + 656·11-s + 206·13-s − 450·15-s + 1.69e3·17-s − 1.36e3·19-s + 4.35e3·21-s − 2.19e3·23-s + 625·25-s + 2.91e3·27-s + 2.21e3·29-s + 1.70e3·31-s − 1.18e4·33-s − 6.05e3·35-s + 846·37-s − 3.70e3·39-s − 1.81e3·41-s + 1.05e4·43-s + 2.02e3·45-s − 1.20e4·47-s + 4.17e4·49-s − 3.04e4·51-s − 3.25e4·53-s + 1.64e4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.86·7-s + 1/3·9-s + 1.63·11-s + 0.338·13-s − 0.516·15-s + 1.41·17-s − 0.866·19-s + 2.15·21-s − 0.866·23-s + 1/5·25-s + 0.769·27-s + 0.489·29-s + 0.317·31-s − 1.88·33-s − 0.834·35-s + 0.101·37-s − 0.390·39-s − 0.168·41-s + 0.868·43-s + 0.149·45-s − 0.797·47-s + 2.48·49-s − 1.63·51-s − 1.59·53-s + 0.731·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 2 p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 + 242 T + p^{5} T^{2} \) |
| 11 | \( 1 - 656 T + p^{5} T^{2} \) |
| 13 | \( 1 - 206 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1690 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1364 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2198 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1700 T + p^{5} T^{2} \) |
| 37 | \( 1 - 846 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1818 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10534 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12074 T + p^{5} T^{2} \) |
| 53 | \( 1 + 32586 T + p^{5} T^{2} \) |
| 59 | \( 1 - 8668 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34670 T + p^{5} T^{2} \) |
| 67 | \( 1 + 47566 T + p^{5} T^{2} \) |
| 71 | \( 1 + 948 T + p^{5} T^{2} \) |
| 73 | \( 1 + 63102 T + p^{5} T^{2} \) |
| 79 | \( 1 + 46536 T + p^{5} T^{2} \) |
| 83 | \( 1 + 88778 T + p^{5} T^{2} \) |
| 89 | \( 1 + 104934 T + p^{5} T^{2} \) |
| 97 | \( 1 + 36254 T + p^{5} T^{2} \) |
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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
−10.17725374368552567446005884750, −9.703086108062566005746241758059, −8.647411502744937884381130655920, −6.94150462852851460269880493442, −6.20944681805223179633840619252, −5.81810057395393420274983185550, −4.18122877122336120612440865362, −3.08078221577963605259660239621, −1.20388656091316167544648215703, 0,
1.20388656091316167544648215703, 3.08078221577963605259660239621, 4.18122877122336120612440865362, 5.81810057395393420274983185550, 6.20944681805223179633840619252, 6.94150462852851460269880493442, 8.647411502744937884381130655920, 9.703086108062566005746241758059, 10.17725374368552567446005884750
