L(s) = 1 | − 4i·5-s − 6i·13-s + 8·17-s − 11·25-s − 4i·29-s + 2i·37-s + 8·41-s − 7·49-s + 4i·53-s − 10i·61-s − 24·65-s − 6·73-s − 32i·85-s − 16·89-s − 18·97-s + ⋯ |
L(s) = 1 | − 1.78i·5-s − 1.66i·13-s + 1.94·17-s − 2.20·25-s − 0.742i·29-s + 0.328i·37-s + 1.24·41-s − 49-s + 0.549i·53-s − 1.28i·61-s − 2.97·65-s − 0.702·73-s − 3.47i·85-s − 1.69·89-s − 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.626733000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.626733000\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 8T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−8.563614314939728499388923405126, −8.008665384019597065619698123983, −7.56723502154780010094923636286, −6.02723024285242528483102462633, −5.49391613207571551538128644678, −4.88167376255002505883363300900, −3.89507888397287736947491483187, −2.91010175768751731444170735721, −1.39901299164329195055601839800, −0.59706028858537316754310186535,
1.60124216691692040003103208613, 2.70787924317443891395253245656, 3.45193035642142507051958864370, 4.28999318051224301124193831558, 5.57748949513655342230201835766, 6.27750501971151210359225889756, 7.09599337490182052157137137928, 7.44561399182354586601083013460, 8.466315049049263157483367972277, 9.548848031972466972528047632167