L(s) = 1 | + (−0.517 + 0.517i)3-s + (−1.73 − 1.73i)5-s + 3.86i·7-s + 2.46i·9-s + (3.34 + 3.34i)11-s + (−0.267 + 0.267i)13-s + 1.79·15-s + 3.46·17-s + (−3.34 + 3.34i)19-s + (−1.99 − 1.99i)21-s − 1.79i·23-s + 0.999i·25-s + (−2.82 − 2.82i)27-s + (−1.73 + 1.73i)29-s − 5.65·31-s + ⋯ |
L(s) = 1 | + (−0.298 + 0.298i)3-s + (−0.774 − 0.774i)5-s + 1.46i·7-s + 0.821i·9-s + (1.00 + 1.00i)11-s + (−0.0743 + 0.0743i)13-s + 0.462·15-s + 0.840·17-s + (−0.767 + 0.767i)19-s + (−0.436 − 0.436i)21-s − 0.373i·23-s + 0.199i·25-s + (−0.544 − 0.544i)27-s + (−0.321 + 0.321i)29-s − 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.704895 + 0.618176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.704895 + 0.618176i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.517 - 0.517i)T - 3iT^{2} \) |
| 5 | \( 1 + (1.73 + 1.73i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.86iT - 7T^{2} \) |
| 11 | \( 1 + (-3.34 - 3.34i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.267 - 0.267i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + (3.34 - 3.34i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.79iT - 23T^{2} \) |
| 29 | \( 1 + (1.73 - 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + (-3.73 - 3.73i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (1.55 + 1.55i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.13 + 5.13i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.73 + 3.73i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4.38 + 4.38i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.79iT - 71T^{2} \) |
| 73 | \( 1 - 2.53iT - 73T^{2} \) |
| 79 | \( 1 + 4.14T + 79T^{2} \) |
| 83 | \( 1 + (-1.55 + 1.55i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−12.28835467634699793999524434610, −11.53298167578633710732111817972, −10.32705734409020857540143590576, −9.223172122317778662003684841480, −8.479490102037169859364761746687, −7.45929893956432491559602336639, −5.95545102531025075458073479136, −4.99173657453981095464944475623, −3.99056227180618458706877151004, −2.04329192091514271459314444020,
0.812210655585464839053756116890, 3.41662618134599008689312620583, 4.05869649589361875992237546028, 5.96613517029465706663440775644, 6.95824600523262478720931181752, 7.53008605783099715376673184886, 8.876059548465936363028140812890, 10.05580543967089846832739309077, 11.18101255064077233367486906277, 11.45412198517818147859551661743