L(s) = 1 | + (0.707 − 0.707i)3-s + (−1.27 − 1.27i)5-s + 0.158i·7-s − 1.00i·9-s + (−3.79 − 3.79i)11-s + (4.21 − 4.21i)13-s − 1.79·15-s + 3.05·17-s + (−2.15 + 2.15i)19-s + (0.112 + 0.112i)21-s − 2.82i·23-s − 1.76i·25-s + (−0.707 − 0.707i)27-s + (−2.09 + 2.09i)29-s − 4.15·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.568 − 0.568i)5-s + 0.0600i·7-s − 0.333i·9-s + (−1.14 − 1.14i)11-s + (1.16 − 1.16i)13-s − 0.464·15-s + 0.740·17-s + (−0.495 + 0.495i)19-s + (0.0245 + 0.0245i)21-s − 0.589i·23-s − 0.353i·25-s + (−0.136 − 0.136i)27-s + (−0.389 + 0.389i)29-s − 0.746·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.862864 - 0.928724i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.862864 - 0.928724i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (1.27 + 1.27i)T + 5iT^{2} \) |
| 7 | \( 1 - 0.158iT - 7T^{2} \) |
| 11 | \( 1 + (3.79 + 3.79i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + (2.15 - 2.15i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (2.09 - 2.09i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + (-5.98 - 5.98i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.60iT - 41T^{2} \) |
| 43 | \( 1 + (-5.75 - 5.75i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (3.55 + 3.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4 - 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.66 - 3.66i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.767 + 0.767i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.317iT - 71T^{2} \) |
| 73 | \( 1 + 1.33iT - 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 + (-0.115 + 0.115i)T - 83iT^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 0.571T + 97T^{2} \) |
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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
−11.00491533468650374821257111830, −10.37880314905146729576214645722, −8.950883994620629193637390458467, −8.137892846510778521745843040446, −7.81753599269061362278311031016, −6.18868490680495513156890602843, −5.37910099614900673517665001160, −3.86161523887856243358622988348, −2.83021223977478145446916601254, −0.837226976059706259047394949791,
2.16088660777696605820702364378, 3.55113341355270170593859663200, 4.45080005763411113761176764510, 5.75886174327497557035177111877, 7.15430293212040656046739945648, 7.71635128749641900934273557370, 8.907635880876023686887265984768, 9.732496570066500375526632267243, 10.77700748297874100083368081954, 11.30332697991790147512699806151