| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 66·5-s − 36·6-s + 176·7-s + 64·8-s + 81·9-s − 264·10-s − 60·11-s − 144·12-s − 658·13-s + 704·14-s + 594·15-s + 256·16-s − 414·17-s + 324·18-s + 956·19-s − 1.05e3·20-s − 1.58e3·21-s − 240·22-s + 600·23-s − 576·24-s + 1.23e3·25-s − 2.63e3·26-s − 729·27-s + 2.81e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.18·5-s − 0.408·6-s + 1.35·7-s + 0.353·8-s + 1/3·9-s − 0.834·10-s − 0.149·11-s − 0.288·12-s − 1.07·13-s + 0.959·14-s + 0.681·15-s + 1/4·16-s − 0.347·17-s + 0.235·18-s + 0.607·19-s − 0.590·20-s − 0.783·21-s − 0.105·22-s + 0.236·23-s − 0.204·24-s + 0.393·25-s − 0.763·26-s − 0.192·27-s + 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.157778415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.157778415\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 66 T + p^{5} T^{2} \) |
| 7 | \( 1 - 176 T + p^{5} T^{2} \) |
| 11 | \( 1 + 60 T + p^{5} T^{2} \) |
| 13 | \( 1 + 658 T + p^{5} T^{2} \) |
| 17 | \( 1 + 414 T + p^{5} T^{2} \) |
| 19 | \( 1 - 956 T + p^{5} T^{2} \) |
| 23 | \( 1 - 600 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 + 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 + 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 - 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 + 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 - 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 + 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 - 166082 T + p^{5} T^{2} \) |
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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
−22.52800198725495672255326514333, −21.08305845822612052852640611502, −19.56932226212139151153669468946, −17.64337214084370917772244375206, −15.87037230282555035666812175867, −14.50668450501514361821022249050, −12.21771055111024451355280485923, −11.12049523748730052768597521933, −7.61949107414732268202119369706, −4.75217914279908534590718045926,
4.75217914279908534590718045926, 7.61949107414732268202119369706, 11.12049523748730052768597521933, 12.21771055111024451355280485923, 14.50668450501514361821022249050, 15.87037230282555035666812175867, 17.64337214084370917772244375206, 19.56932226212139151153669468946, 21.08305845822612052852640611502, 22.52800198725495672255326514333
