L(s) = 1 | + (1.27 + 1.27i)5-s − 0.158i·7-s + (−3.79 − 3.79i)11-s + (4.21 − 4.21i)13-s − 3.05·17-s + (2.15 − 2.15i)19-s − 2.82i·23-s − 1.76i·25-s + (2.09 − 2.09i)29-s + 4.15·31-s + (0.202 − 0.202i)35-s + (5.98 + 5.98i)37-s + 2.60i·41-s + (−5.75 − 5.75i)43-s + 2.82·47-s + ⋯ |
L(s) = 1 | + (0.568 + 0.568i)5-s − 0.0600i·7-s + (−1.14 − 1.14i)11-s + (1.16 − 1.16i)13-s − 0.740·17-s + (0.495 − 0.495i)19-s − 0.589i·23-s − 0.353i·25-s + (0.389 − 0.389i)29-s + 0.746·31-s + (0.0341 − 0.0341i)35-s + (0.984 + 0.984i)37-s + 0.406i·41-s + (−0.877 − 0.877i)43-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.647378672\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647378672\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
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
−9.929957133334960655631520040875, −8.583696456383401055668344073459, −8.297936488230250524695889609121, −7.16010065806566852756915793738, −6.12086004708035417279065233967, −5.70702213291210347592855966887, −4.51938842313453371905002371158, −3.15145762452195230967690103124, −2.57725482442369525225877465647, −0.76117036173217255813321407855,
1.44366313949828988338625345765, 2.41424301465507757012953073649, 3.88242241308189720045338658096, 4.80863875279165426220078198403, 5.60122724978862802237608301527, 6.55334382286674438643856962267, 7.43663857675278653381506038898, 8.362363318454017895516478042568, 9.190604798765203406435762661394, 9.764567490054182329347449977037