| L(s) = 1 | + 9·3-s + 46·5-s − 148·7-s + 81·9-s − 121·11-s + 574·13-s + 414·15-s − 722·17-s − 2.16e3·19-s − 1.33e3·21-s + 2.53e3·23-s − 1.00e3·25-s + 729·27-s + 4.65e3·29-s − 5.03e3·31-s − 1.08e3·33-s − 6.80e3·35-s + 8.11e3·37-s + 5.16e3·39-s − 5.13e3·41-s − 8.30e3·43-s + 3.72e3·45-s − 2.47e4·47-s + 5.09e3·49-s − 6.49e3·51-s − 2.87e4·53-s − 5.56e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.822·5-s − 1.14·7-s + 1/3·9-s − 0.301·11-s + 0.942·13-s + 0.475·15-s − 0.605·17-s − 1.37·19-s − 0.659·21-s + 0.999·23-s − 0.322·25-s + 0.192·27-s + 1.02·29-s − 0.940·31-s − 0.174·33-s − 0.939·35-s + 0.974·37-s + 0.543·39-s − 0.477·41-s − 0.684·43-s + 0.274·45-s − 1.63·47-s + 0.303·49-s − 0.349·51-s − 1.40·53-s − 0.248·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 46 T + p^{5} T^{2} \) |
| 7 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 - 574 T + p^{5} T^{2} \) |
| 17 | \( 1 + 722 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2536 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4650 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5032 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8118 T + p^{5} T^{2} \) |
| 41 | \( 1 + 5138 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8304 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24728 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28746 T + p^{5} T^{2} \) |
| 59 | \( 1 - 5860 T + p^{5} T^{2} \) |
| 61 | \( 1 + 53658 T + p^{5} T^{2} \) |
| 67 | \( 1 + 30908 T + p^{5} T^{2} \) |
| 71 | \( 1 - 69648 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18446 T + p^{5} T^{2} \) |
| 79 | \( 1 - 25300 T + p^{5} T^{2} \) |
| 83 | \( 1 - 17556 T + p^{5} T^{2} \) |
| 89 | \( 1 - 132570 T + p^{5} T^{2} \) |
| 97 | \( 1 - 70658 T + p^{5} T^{2} \) |
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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
−9.512501667024436289263596636406, −8.903091918305836916705810823152, −7.967605201118617399647396061314, −6.55612334373422905983144506636, −6.26991354954120317844394216143, −4.85978215829857954091365933762, −3.61631466157421447920430945641, −2.68479057891366900259162671218, −1.58224138048241172009957329448, 0,
1.58224138048241172009957329448, 2.68479057891366900259162671218, 3.61631466157421447920430945641, 4.85978215829857954091365933762, 6.26991354954120317844394216143, 6.55612334373422905983144506636, 7.967605201118617399647396061314, 8.903091918305836916705810823152, 9.512501667024436289263596636406
