| L(s) = 1 | − 4.32i·2-s + (−14.6 + 22.6i)3-s + 45.3·4-s − 48.7i·5-s + (97.8 + 63.4i)6-s − 61.9·7-s − 472. i·8-s + (−297. − 665. i)9-s − 210.·10-s + 1.34e3i·11-s + (−666. + 1.02e3i)12-s + 1.84e3·13-s + 267. i·14-s + (1.10e3 + 716. i)15-s + 861.·16-s − 5.40e3i·17-s + ⋯ |
| L(s) = 1 | − 0.540i·2-s + (−0.544 + 0.838i)3-s + 0.708·4-s − 0.389i·5-s + (0.453 + 0.293i)6-s − 0.180·7-s − 0.922i·8-s + (−0.407 − 0.913i)9-s − 0.210·10-s + 1.01i·11-s + (−0.385 + 0.594i)12-s + 0.840·13-s + 0.0975i·14-s + (0.327 + 0.212i)15-s + 0.210·16-s − 1.09i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.82230 - 0.539259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82230 - 0.539259i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (14.6 - 22.6i)T \) |
| 23 | \( 1 - 2.53e3iT \) |
| good | 2 | \( 1 + 4.32iT - 64T^{2} \) |
| 5 | \( 1 + 48.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 61.9T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.84e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 5.40e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.04e4T + 4.70e7T^{2} \) |
| 29 | \( 1 + 3.00e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 537.T + 8.87e8T^{2} \) |
| 37 | \( 1 - 5.98e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.97e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.03e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 4.73e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 7.73e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.17e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.11e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.40e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.46e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.27e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 4.73e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 9.51e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.24e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 9.03e5T + 8.32e11T^{2} \) |
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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
−13.11928555449755577167077282361, −11.92774957399851318165036053984, −11.31726662537738955080733031141, −10.06301825680394416221488621582, −9.315190399152012553712727343916, −7.35821687867214398354644006095, −5.96325277510527322223377122512, −4.50002812630985356306172769072, −2.99726227642146665404082191211, −0.976636899628983750622002272688,
1.24675506615850790707124741686, 3.02853377586161468564991722099, 5.59801009332672293187466221637, 6.38804172851661570025889849807, 7.45548344374801189168691582600, 8.558910815109926116156389773524, 10.69944706564803783906835241433, 11.28994789261618366648330385004, 12.48200753830998760475825741468, 13.68829722045968541143995704257
