| L(s) = 1 | + 3.69i·3-s − 0.634i·5-s + (1.82 + 6.75i)7-s − 4.65·9-s − 13.3·11-s + 21.5i·13-s + 2.34·15-s − 6.12i·17-s − 19.7i·19-s + (−24.9 + 6.75i)21-s + 28.6·23-s + 24.5·25-s + 16.0i·27-s + 37.3·29-s − 20.9i·31-s + ⋯ |
| L(s) = 1 | + 1.23i·3-s − 0.126i·5-s + (0.261 + 0.965i)7-s − 0.517·9-s − 1.21·11-s + 1.65i·13-s + 0.156·15-s − 0.360i·17-s − 1.03i·19-s + (−1.18 + 0.321i)21-s + 1.24·23-s + 0.983·25-s + 0.594i·27-s + 1.28·29-s − 0.674i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.781700 + 1.02134i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.781700 + 1.02134i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1.82 - 6.75i)T \) |
| good | 3 | \( 1 - 3.69iT - 9T^{2} \) |
| 5 | \( 1 + 0.634iT - 25T^{2} \) |
| 11 | \( 1 + 13.3T + 121T^{2} \) |
| 13 | \( 1 - 21.5iT - 169T^{2} \) |
| 17 | \( 1 + 6.12iT - 289T^{2} \) |
| 19 | \( 1 + 19.7iT - 361T^{2} \) |
| 23 | \( 1 - 28.6T + 529T^{2} \) |
| 29 | \( 1 - 37.3T + 841T^{2} \) |
| 31 | \( 1 + 20.9iT - 961T^{2} \) |
| 37 | \( 1 + 23.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 70.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 46T + 1.84e3T^{2} \) |
| 47 | \( 1 + 1.05iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 15.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 14.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 86.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 24.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 45.0T + 5.04e3T^{2} \) |
| 73 | \( 1 - 51.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 33.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 77.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 90.1iT - 9.40e3T^{2} \) |
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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
−13.83071481423856468849927331678, −12.56547903522165021603064846127, −11.39489274290059803434798135476, −10.56039047735986703431841969074, −9.287537656163916890543321720564, −8.749809490565892558667468858605, −7.00601625419374572933020896870, −5.28409833428733073606485485883, −4.51325613682464802761382577211, −2.67831962312592918901404247959,
1.01899414553256883267220382943, 2.99685551692823283492648391381, 5.06773631757479520089011390328, 6.53486111656790312554094420983, 7.66156821726362119423824731426, 8.199617576279716450550893781928, 10.24370457309726145034331031347, 10.83954611935714606543271867910, 12.48743493288576484804537512624, 12.94449416704269444101920763591
