| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 1.58i·5-s − 5.24·7-s − 2.82i·8-s − 2.24·10-s + 5.82i·11-s − 7.41i·14-s + 4.00·16-s − 3.17i·20-s − 8.24·22-s + 2.48·25-s + 10.4·28-s + 2.82i·29-s + 0.757·31-s + 5.65i·32-s + ⋯ |
| L(s) = 1 | + 0.999i·2-s − 1.00·4-s + 0.709i·5-s − 1.98·7-s − 1.00i·8-s − 0.709·10-s + 1.75i·11-s − 1.98i·14-s + 1.00·16-s − 0.709i·20-s − 1.75·22-s + 0.497·25-s + 1.98·28-s + 0.525i·29-s + 0.136·31-s + 1.00i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.659991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.659991i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.58iT - 5T^{2} \) |
| 7 | \( 1 + 5.24T + 7T^{2} \) |
| 11 | \( 1 - 5.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 0.757T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−12.83555349841965283620104943401, −12.30748800919841413833573223916, −10.39320881627126453535271634121, −9.798770639711175160629424131398, −8.966689318783669584270217950417, −7.29094383179781281497956501507, −6.88077449914203091387254420731, −5.89356311258118591278667320640, −4.35202359523443786773244670771, −3.02891429392479997191239342200,
0.55514023329223631181035262769, 2.92247024654684937327965633141, 3.80586137743114081422903519392, 5.43491966857946917100599255991, 6.45991083049453539210012708206, 8.293053947003363519032759692182, 9.105680329696175831997801724570, 9.868296806403300730550566256931, 10.87612996371669371434773784733, 11.92051130217156658328379194318
