| L(s) = 1 | − 1.73i·3-s + 1.73i·5-s − 2.99·9-s − 3·11-s + 6.92i·13-s + 2.99·15-s + 17.3i·17-s + 10.3i·19-s + 36·23-s + 22·25-s + 5.19i·27-s − 51·29-s + 12.1i·31-s + 5.19i·33-s + 22·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.346i·5-s − 0.333·9-s − 0.272·11-s + 0.532i·13-s + 0.199·15-s + 1.01i·17-s + 0.546i·19-s + 1.56·23-s + 0.880·25-s + 0.192i·27-s − 1.75·29-s + 0.391i·31-s + 0.157i·33-s + 0.594·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.507385887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507385887\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.73iT - 25T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 - 6.92iT - 169T^{2} \) |
| 17 | \( 1 - 17.3iT - 289T^{2} \) |
| 19 | \( 1 - 10.3iT - 361T^{2} \) |
| 23 | \( 1 - 36T + 529T^{2} \) |
| 29 | \( 1 + 51T + 841T^{2} \) |
| 31 | \( 1 - 12.1iT - 961T^{2} \) |
| 37 | \( 1 - 22T + 1.36e3T^{2} \) |
| 41 | \( 1 - 24.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10T + 1.84e3T^{2} \) |
| 47 | \( 1 - 90.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51T + 2.80e3T^{2} \) |
| 59 | \( 1 + 74.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 69.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 68T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.04e3T^{2} \) |
| 73 | \( 1 - 20.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 125T + 6.24e3T^{2} \) |
| 83 | \( 1 - 154. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 72.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 147. iT - 9.40e3T^{2} \) |
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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
−10.88484581889301396937773275002, −9.680494070722267451736399999412, −8.805827613920809630482659868080, −7.86420281077274157589746719140, −7.02359611513329886255111438633, −6.20223034713417550531271925703, −5.16439083935614216247502912046, −3.84082722727212344194645525780, −2.64447887327741477051064543753, −1.32700219554141377948188496052,
0.61990055605591198295575321979, 2.53134148614551778941647434480, 3.64638313430677821621083119574, 4.93022628407194996076895031433, 5.44368997075712284515016128735, 6.85019895625701198123414534850, 7.70964950945993624050986492244, 8.900985989644728305812018539998, 9.330449821790165089067214546887, 10.43457411803844137389744394665
