| L(s) = 1 | − 1.64·3-s − 4.56i·5-s − 6.28·9-s + 12.3·11-s + 18.3i·13-s + 7.52i·15-s − 13.0·17-s + 3.02·19-s − 30.3i·23-s + 4.19·25-s + 25.1·27-s − 22.7i·29-s − 22.5i·31-s − 20.4·33-s − 13.7i·37-s + ⋯ |
| L(s) = 1 | − 0.549·3-s − 0.912i·5-s − 0.697·9-s + 1.12·11-s + 1.41i·13-s + 0.501i·15-s − 0.766·17-s + 0.159·19-s − 1.31i·23-s + 0.167·25-s + 0.933·27-s − 0.785i·29-s − 0.727i·31-s − 0.618·33-s − 0.372i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1091827712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1091827712\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.64T + 9T^{2} \) |
| 5 | \( 1 + 4.56iT - 25T^{2} \) |
| 11 | \( 1 - 12.3T + 121T^{2} \) |
| 13 | \( 1 - 18.3iT - 169T^{2} \) |
| 17 | \( 1 + 13.0T + 289T^{2} \) |
| 19 | \( 1 - 3.02T + 361T^{2} \) |
| 23 | \( 1 + 30.3iT - 529T^{2} \) |
| 29 | \( 1 + 22.7iT - 841T^{2} \) |
| 31 | \( 1 + 22.5iT - 961T^{2} \) |
| 37 | \( 1 + 13.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 60.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 39.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 20.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.76iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 11.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 108. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 79.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 12.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 98.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 131. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 28.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 157.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.6T + 9.40e3T^{2} \) |
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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
−8.805026617368851985891277613029, −8.320904227289890964171490077305, −6.85626177422677864424520246777, −6.47465719176295220553330890766, −5.47669925286243154232462414022, −4.56906004456172627400188711683, −3.98814278897776399326791184466, −2.44532428151094038680796686340, −1.26865614622690999948953859674, −0.03399921346840443193693006631,
1.45710341540525153200084917968, 2.97692834705609643024496518722, 3.48783288254759194113278075604, 4.90841881908842858830325459658, 5.64445845183538177846461484254, 6.53939132878930402900131963113, 7.02291793622311614933473656649, 8.124556742362631746979681642299, 8.853508179513577078711614247567, 9.808065696354190011858351546174
