L(s) = 1 | + (−0.707 − 0.707i)3-s + (−2.68 + 2.68i)5-s + 2.15i·7-s + 1.00i·9-s + (−1.79 + 1.79i)11-s + (1.38 + 1.38i)13-s + 3.79·15-s − 0.224·17-s + (−0.158 − 0.158i)19-s + (1.52 − 1.52i)21-s + 2.82i·23-s − 9.42i·25-s + (0.707 − 0.707i)27-s + (−1.85 − 1.85i)29-s − 1.84·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−1.20 + 1.20i)5-s + 0.816i·7-s + 0.333i·9-s + (−0.542 + 0.542i)11-s + (0.383 + 0.383i)13-s + 0.980·15-s − 0.0545·17-s + (−0.0364 − 0.0364i)19-s + (0.333 − 0.333i)21-s + 0.589i·23-s − 1.88i·25-s + (0.136 − 0.136i)27-s + (−0.344 − 0.344i)29-s − 0.330·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.403582 + 0.517295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.403582 + 0.517295i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.68 - 2.68i)T - 5iT^{2} \) |
| 7 | \( 1 - 2.15iT - 7T^{2} \) |
| 11 | \( 1 + (1.79 - 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \) |
| 17 | \( 1 + 0.224T + 17T^{2} \) |
| 19 | \( 1 + (0.158 + 0.158i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (1.85 + 1.85i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + (3.66 - 3.66i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.88iT - 41T^{2} \) |
| 43 | \( 1 + (-7.75 + 7.75i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-7.51 + 7.51i)T - 53iT^{2} \) |
| 59 | \( 1 + (4 - 4i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.98 - 5.98i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.4 - 10.4i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.31iT - 71T^{2} \) |
| 73 | \( 1 - 5.97iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.42iT - 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−12.56182976436496359582323697310, −11.72945997045031270534714976895, −11.13521006977461037934171873580, −10.09893592754649302847579653094, −8.610806199054071442144141168432, −7.53774122990806353898331396997, −6.80369952507907483445181183668, −5.54023379364062393409188443845, −3.94922946611395982170734112672, −2.50237359061151910810819593173,
0.61656417361522987083582849567, 3.63285978461310777062656835680, 4.54479900243807865214689074709, 5.65306215981697949756349832348, 7.30051699056076270324603199956, 8.234146899399886622262802215521, 9.130621642284037051286281439428, 10.55868969496211497346224533341, 11.18333869710824321466027648954, 12.31145730847558914635624403088