L(s) = 1 | + 2-s + 4-s + (0.707 + 0.707i)7-s + 8-s + (−0.707 − 1.70i)11-s + (0.707 + 0.707i)14-s + 16-s + (−0.707 − 1.70i)22-s + (−0.707 + 0.707i)25-s + (0.707 + 0.707i)28-s + (0.707 + 0.292i)29-s + 32-s + (−1.70 + 0.707i)37-s + (0.707 − 0.292i)43-s + (−0.707 − 1.70i)44-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + (0.707 + 0.707i)7-s + 8-s + (−0.707 − 1.70i)11-s + (0.707 + 0.707i)14-s + 16-s + (−0.707 − 1.70i)22-s + (−0.707 + 0.707i)25-s + (0.707 + 0.707i)28-s + (0.707 + 0.292i)29-s + 32-s + (−1.70 + 0.707i)37-s + (0.707 − 0.292i)43-s + (−0.707 − 1.70i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.305149580\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305149580\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (1 - i)T - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.204966415423411949831640764812, −8.354207067599157045508583900451, −7.83513959673119074962227316449, −6.80169627582260073126496507196, −5.81845693527872974969857032615, −5.45805981228109190225702640315, −4.57702720483492046667821306302, −3.42473850030887990049724625216, −2.75556111537211227883954330145, −1.56054297437269078064248583873,
1.66567673276993566850042753003, 2.48715432468361199316269978137, 3.77034675330082694232536658859, 4.58067041896896796114031429709, 5.03613554830650975941025802821, 6.09717798310453723080261266767, 7.07022180961208813909460484335, 7.53225370569196250488380606234, 8.247942864182657109123441603797, 9.546266029730766980308576644004