| L(s) = 1 | + 5.65·2-s + 15.5·3-s + 32.0·4-s + 37.6i·5-s + 88.1·6-s + 272. i·7-s + 181.·8-s + 243·9-s + 212. i·10-s − 1.50e3i·11-s + 498.·12-s + 2.82e3·13-s + 1.54e3i·14-s + 586. i·15-s + 1.02e3·16-s + 7.22e3i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.500·4-s + 0.301i·5-s + 0.408·6-s + 0.795i·7-s + 0.353·8-s + 0.333·9-s + 0.212i·10-s − 1.13i·11-s + 0.288·12-s + 1.28·13-s + 0.562i·14-s + 0.173i·15-s + 0.250·16-s + 1.47i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.084967996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.084967996\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (9.18e3 - 7.97e3i)T \) |
| good | 5 | \( 1 - 37.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 272. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.50e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.82e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.22e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 4.05e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 2.16e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.31e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.07e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 2.39e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 5.68e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 6.56e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 7.99e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.76e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.28e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.53e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + 3.55e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.62e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.81e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.28e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 4.56e5iT - 8.32e11T^{2} \) |
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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
−12.32439746583490229253330105298, −11.22991046274828858391142734537, −10.31818939380591710553676395091, −8.753417582208785736749391949968, −8.110048261662500978755089989733, −6.41930627761043494141116920204, −5.69069296533011707308472897046, −3.92807001081297448110691005114, −3.01619706867625052398872132788, −1.53010401031963994314785951486,
1.03409121972244938409382684786, 2.60497799540810507181614655452, 4.00843505646835351185856300529, 4.90979864560986991752033043460, 6.59172035565666104811060039025, 7.47919432186354611590732755525, 8.748566663554604684713380907076, 9.935062637587889567132725991944, 10.98115075510637985269014214929, 12.15998382882236549828835547859
