| L(s) = 1 | + 7.57i·5-s − 9.11·7-s − 0.442i·11-s − 11.1·13-s + 8.01i·17-s + 8.11·19-s − 23.6i·23-s − 32.3·25-s + 53.0i·29-s − 29.3·31-s − 69.0i·35-s + 18.4·37-s − 44.9i·41-s − 23·43-s − 8.45i·47-s + ⋯ |
| L(s) = 1 | + 1.51i·5-s − 1.30·7-s − 0.0402i·11-s − 0.855·13-s + 0.471i·17-s + 0.427·19-s − 1.02i·23-s − 1.29·25-s + 1.82i·29-s − 0.946·31-s − 1.97i·35-s + 0.499·37-s − 1.09i·41-s − 0.534·43-s − 0.179i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3280555013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3280555013\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 7.57iT - 25T^{2} \) |
| 7 | \( 1 + 9.11T + 49T^{2} \) |
| 11 | \( 1 + 0.442iT - 121T^{2} \) |
| 13 | \( 1 + 11.1T + 169T^{2} \) |
| 17 | \( 1 - 8.01iT - 289T^{2} \) |
| 19 | \( 1 - 8.11T + 361T^{2} \) |
| 23 | \( 1 + 23.6iT - 529T^{2} \) |
| 29 | \( 1 - 53.0iT - 841T^{2} \) |
| 31 | \( 1 + 29.3T + 961T^{2} \) |
| 37 | \( 1 - 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 44.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.45iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 76.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.35T + 3.72e3T^{2} \) |
| 67 | \( 1 - 109.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.35T + 5.32e3T^{2} \) |
| 79 | \( 1 + 1.58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 8.45iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
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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
−9.492810053114104837580049893425, −8.478373464382701023134647247420, −7.26182469867451557852843343302, −6.88239795348538787731692413705, −6.17837467347544906162815881345, −5.13310003067069183370205483573, −3.68093268152435895974915474413, −3.12215305173925161637741774230, −2.17203933537584032254208553242, −0.10777303450648228405966075237,
1.00758716438021958041586962291, 2.46751844020216983894413042182, 3.64846035551752519182243812463, 4.61800685748822399942637657817, 5.43785400791810356999895704057, 6.23498650179582677603326229138, 7.34234954492525969748594137686, 8.029801049417491199668804065091, 9.155574861263818715542222245629, 9.506052626722110316475322066014
