| L(s) = 1 | + 3.92i·5-s − 12.7·7-s + 16.5i·11-s − 2.79·13-s − 2.54i·17-s − 21.5·19-s + 3.00i·23-s + 9.59·25-s − 15.2i·29-s + 27.2·31-s − 50.2i·35-s − 10.4·37-s + 39.8i·41-s − 34.1·43-s − 67.2i·47-s + ⋯ |
| L(s) = 1 | + 0.784i·5-s − 1.82·7-s + 1.50i·11-s − 0.215·13-s − 0.149i·17-s − 1.13·19-s + 0.130i·23-s + 0.383·25-s − 0.525i·29-s + 0.877·31-s − 1.43i·35-s − 0.281·37-s + 0.971i·41-s − 0.795·43-s − 1.42i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3457257762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3457257762\) |
| \(L(2)\) |
|
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 | \( 1 - 3.92iT - 25T^{2} \) |
| 7 | \( 1 + 12.7T + 49T^{2} \) |
| 11 | \( 1 - 16.5iT - 121T^{2} \) |
| 13 | \( 1 + 2.79T + 169T^{2} \) |
| 17 | \( 1 + 2.54iT - 289T^{2} \) |
| 19 | \( 1 + 21.5T + 361T^{2} \) |
| 23 | \( 1 - 3.00iT - 529T^{2} \) |
| 29 | \( 1 + 15.2iT - 841T^{2} \) |
| 31 | \( 1 - 27.2T + 961T^{2} \) |
| 37 | \( 1 + 10.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 100. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.11iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 108.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 52.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 25.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 60.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.62iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101.T + 9.40e3T^{2} \) |
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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
−9.550396195820424039004462398980, −8.521120299635969259753487039356, −7.35969077640947265815810528618, −6.67543239557272494501443171817, −6.33348948318832809942377681675, −4.97366428174343854602036541180, −3.92184119677778897148074726239, −3.00034452972367495357682889141, −2.13322889474414532635529366781, −0.12091415722327326370214049736,
0.925066775519780114120343428531, 2.69167309276905656693169027382, 3.47986473447349207892670291737, 4.47804489796158475195839637509, 5.69859422049306185223629015088, 6.26178478942014697925065840742, 7.05334962600671966087001205126, 8.342689434289112471877447360704, 8.838719610004290378332112398944, 9.547008924570314246847135602842
