L(s) = 1 | + 14.5i·2-s − 15.5·3-s − 148.·4-s + 0.687·5-s − 227. i·6-s − 18.0i·7-s − 1.23e3i·8-s + 243·9-s + 10.0i·10-s + (−756. − 1.09e3i)11-s + 2.32e3·12-s + 2.82e3i·13-s + 262.·14-s − 10.7·15-s + 8.55e3·16-s − 2.97e3i·17-s + ⋯ |
L(s) = 1 | + 1.82i·2-s − 0.577·3-s − 2.32·4-s + 0.00550·5-s − 1.05i·6-s − 0.0525i·7-s − 2.42i·8-s + 0.333·9-s + 0.0100i·10-s + (−0.568 − 0.822i)11-s + 1.34·12-s + 1.28i·13-s + 0.0957·14-s − 0.00317·15-s + 2.08·16-s − 0.605i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.000749452 - 0.000393182i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.000749452 - 0.000393182i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 15.5T \) |
| 11 | \( 1 + (756. + 1.09e3i)T \) |
good | 2 | \( 1 - 14.5iT - 64T^{2} \) |
| 5 | \( 1 - 0.687T + 1.56e4T^{2} \) |
| 7 | \( 1 + 18.0iT - 1.17e5T^{2} \) |
| 13 | \( 1 - 2.82e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.97e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 8.40e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.97e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 8.49e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.26e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 8.83e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.41e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.31e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 7.55e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.53e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.03e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.41e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.63e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 3.04e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 5.97e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 2.22e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 7.18e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.24e6T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.32e11T^{2} \) |
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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
−15.72771226203447015029561428538, −14.15583594534646889812125596934, −13.41525904248760842225405369574, −11.59847183646276988873366365997, −9.691798073277909240586560590191, −8.320861504384774973210894329732, −6.96783082834047170789225713804, −5.85739221170630965555202580293, −4.49072428382574826542811934492, −0.00047192508862913396893149720,
1.89791084913676605870523943980, 3.79639424741695901015245695918, 5.44734230132573887943548551328, 8.132072795981122699137162484179, 10.03039556925947035704789576998, 10.47744616120847609179437041437, 12.03742720556405985836736222290, 12.58365694074713861010791577326, 13.86765345191041964595666888313, 15.54572151025151120271105700514