| L(s) = 1 | − 3.34i·5-s + 4.73i·7-s − 4.24·11-s + 13-s + 3.34i·17-s + 1.26i·19-s − 7.34·23-s − 6.19·25-s − 4.00i·29-s + 6i·31-s + 15.8·35-s − 9.19·37-s + 7.34i·41-s − 2.19i·43-s + 3.10·47-s + ⋯ |
| L(s) = 1 | − 1.49i·5-s + 1.78i·7-s − 1.27·11-s + 0.277·13-s + 0.811i·17-s + 0.290i·19-s − 1.53·23-s − 1.23·25-s − 0.743i·29-s + 1.07i·31-s + 2.67·35-s − 1.51·37-s + 1.14i·41-s − 0.334i·43-s + 0.453·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6656304341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6656304341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.34iT - 5T^{2} \) |
| 7 | \( 1 - 4.73iT - 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.34iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 + 7.34T + 23T^{2} \) |
| 29 | \( 1 + 4.00iT - 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 - 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 - 3.10T + 47T^{2} \) |
| 53 | \( 1 - 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 3.10T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 - 7.26iT - 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 1.19T + 73T^{2} \) |
| 79 | \( 1 - 1.26iT - 79T^{2} \) |
| 83 | \( 1 + 8.48T + 83T^{2} \) |
| 89 | \( 1 - 9.38iT - 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.768426622984803575270625183453, −8.945491003304172915759356482390, −8.320862944997429657332229518967, −7.980913092092430831347741392821, −6.34277977455457696297590820738, −5.49524878315876078792839930413, −5.17049697200089817464119947230, −3.99557070560224783081753651921, −2.59928521179697943938234306164, −1.63853021297597360293966384768,
0.26070986207376858384835213848, 2.15980085712322007580241192679, 3.26698610727380333379902549762, 3.97339442783899826064389072727, 5.13663160437584494279518020258, 6.27192997719998043433316941762, 7.16873320982124097375346153950, 7.43199360448542133495681766250, 8.339353813809505959018620233823, 9.841210384080524724826832626538
