L(s) = 1 | + 2.64i·7-s − 4i·11-s + 8i·23-s + 5·25-s − 10.5·29-s + 6·37-s + 5.29i·43-s − 7.00·49-s + 10.5·53-s + 15.8i·67-s − 16i·71-s + 10.5·77-s + 15.8i·79-s + 20i·107-s − 18·109-s + ⋯ |
L(s) = 1 | + 0.999i·7-s − 1.20i·11-s + 1.66i·23-s + 25-s − 1.96·29-s + 0.986·37-s + 0.806i·43-s − 49-s + 1.45·53-s + 1.93i·67-s − 1.89i·71-s + 1.20·77-s + 1.78i·79-s + 1.93i·107-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.428927337\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.428927337\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.718782811869008928280753253005, −7.935806734056698757995056245383, −7.24983385463126012597621684221, −6.23175803978641084918528897380, −5.66111272808665735256117518001, −5.12423103226544268649342374279, −3.89767791402347863241630828830, −3.18181539933765717095312861824, −2.30020590929615242346956981262, −1.13469035014923703245647784304,
0.44227023936944190438865429585, 1.70463352289746863237918581525, 2.65553400733955010443916488962, 3.80985509520358652365013434177, 4.41307807814690098082746382415, 5.12305601928994183347257562702, 6.15700409157034182000590658016, 7.00654267519237419057726562349, 7.34233595291588432134201943112, 8.194943617996727042326131941446