L(s) = 1 | + (−1.96 − 1.96i)3-s + 0.264i·5-s + (0.707 + 0.707i)7-s + 4.69i·9-s + (1.34 + 1.34i)11-s + (−2.27 − 2.27i)13-s + (0.519 − 0.519i)15-s + (0.388 − 0.388i)17-s + (1.69 − 1.69i)19-s − 2.77i·21-s + 8.48·23-s + 4.92·25-s + (3.33 − 3.33i)27-s + (−4.81 − 4.81i)29-s + 2.85·31-s + ⋯ |
L(s) = 1 | + (−1.13 − 1.13i)3-s + 0.118i·5-s + (0.267 + 0.267i)7-s + 1.56i·9-s + (0.406 + 0.406i)11-s + (−0.629 − 0.629i)13-s + (0.134 − 0.134i)15-s + (0.0941 − 0.0941i)17-s + (0.389 − 0.389i)19-s − 0.605i·21-s + 1.76·23-s + 0.985·25-s + (0.641 − 0.641i)27-s + (−0.894 − 0.894i)29-s + 0.512·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9868420915\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9868420915\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (1.29 + 6.27i)T \) |
good | 3 | \( 1 + (1.96 + 1.96i)T + 3iT^{2} \) |
| 5 | \( 1 - 0.264iT - 5T^{2} \) |
| 11 | \( 1 + (-1.34 - 1.34i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.27 + 2.27i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.388 + 0.388i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.69 + 1.69i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + (4.81 + 4.81i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 43 | \( 1 - 4.60iT - 43T^{2} \) |
| 47 | \( 1 + (-4.98 + 4.98i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.45 + 1.45i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 + 7.94iT - 61T^{2} \) |
| 67 | \( 1 + (-2.37 + 2.37i)T - 67iT^{2} \) |
| 71 | \( 1 + (10.1 + 10.1i)T + 71iT^{2} \) |
| 73 | \( 1 + 4.71iT - 73T^{2} \) |
| 79 | \( 1 + (-2.09 - 2.09i)T + 79iT^{2} \) |
| 83 | \( 1 - 5.22T + 83T^{2} \) |
| 89 | \( 1 + (8.76 + 8.76i)T + 89iT^{2} \) |
| 97 | \( 1 + (-1.75 + 1.75i)T - 97iT^{2} \) |
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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
−9.592981744468666619053270565806, −8.674274136266823493239263290713, −7.51436580748363222855969086034, −7.12568214752546434489895391566, −6.27842297135186902757765938922, −5.34860583195494977838045758601, −4.76219091856951205454929026727, −3.06674687855330956507930677082, −1.82613097455972437271740911379, −0.59730238722544375337206945198,
1.16164015866505977995870036339, 3.11451378307140158113392950986, 4.17466887315521591245069903208, 4.93321108644575530541452621963, 5.54736866928595491369397753487, 6.61173606220496994228588663738, 7.35250431651174217388466494331, 8.745413309451071678988695288120, 9.303484051135791019231579421479, 10.18595716818332195194460072261