L(s) = 1 | + 1.02i·3-s − 3.54i·5-s − 0.166i·7-s + 1.95·9-s + 0.452i·11-s + 3.43·13-s + 3.62·15-s + (4.12 − 0.0679i)17-s + 7.68·19-s + 0.170·21-s + 8.47i·23-s − 7.54·25-s + 5.06i·27-s + 1.93i·29-s + 2.57i·31-s + ⋯ |
L(s) = 1 | + 0.591i·3-s − 1.58i·5-s − 0.0630i·7-s + 0.650·9-s + 0.136i·11-s + 0.953·13-s + 0.936·15-s + (0.999 − 0.0164i)17-s + 1.76·19-s + 0.0372·21-s + 1.76i·23-s − 1.50·25-s + 0.975i·27-s + 0.358i·29-s + 0.462i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.398785867\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398785867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (-4.12 + 0.0679i)T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 1.02iT - 3T^{2} \) |
| 5 | \( 1 + 3.54iT - 5T^{2} \) |
| 7 | \( 1 + 0.166iT - 7T^{2} \) |
| 11 | \( 1 - 0.452iT - 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 - 8.47iT - 23T^{2} \) |
| 29 | \( 1 - 1.93iT - 29T^{2} \) |
| 31 | \( 1 - 2.57iT - 31T^{2} \) |
| 37 | \( 1 + 1.31iT - 37T^{2} \) |
| 41 | \( 1 + 4.56iT - 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 61 | \( 1 - 8.42iT - 61T^{2} \) |
| 67 | \( 1 + 8.66T + 67T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 4.18iT - 73T^{2} \) |
| 79 | \( 1 + 12.9iT - 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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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
−8.720927657193050668063855229128, −7.64297208120553245140744595010, −7.29817549470610870746246427759, −5.90235518164442338299321642508, −5.29987394484995046188485640066, −4.83125154199338055558239993369, −3.80786419037960770282707215008, −3.35870133706457481635447423094, −1.51423509360801408501593860773, −1.08787758377481316812267074626,
0.898697572760017353979044946002, 2.00945326715582763010838273657, 3.03942270599557509042579364028, 3.52506003095246868762568896114, 4.63800247991080533893423225671, 5.76452638293795802162639549385, 6.43316797279964341626046230796, 6.88750443340702167788151269026, 7.71724713507689940898290777889, 8.073743348179927550851865335980