L(s) = 1 | + (0.621 + 1.27i)2-s + (−1.22 + 1.57i)4-s + 3.69i·5-s + 7-s + (−2.76 − 0.578i)8-s + (−4.69 + 2.29i)10-s + 3.21i·11-s − 5.08i·13-s + (0.621 + 1.27i)14-s + (−0.985 − 3.87i)16-s − 0.616·17-s + 4.48i·19-s + (−5.83 − 4.54i)20-s + (−4.08 + 1.99i)22-s − 1.38·23-s + ⋯ |
L(s) = 1 | + (0.439 + 0.898i)2-s + (−0.613 + 0.789i)4-s + 1.65i·5-s + 0.377·7-s + (−0.978 − 0.204i)8-s + (−1.48 + 0.726i)10-s + 0.968i·11-s − 1.40i·13-s + (0.166 + 0.339i)14-s + (−0.246 − 0.969i)16-s − 0.149·17-s + 1.02i·19-s + (−1.30 − 1.01i)20-s + (−0.870 + 0.425i)22-s − 0.288·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.156425 + 1.51270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156425 + 1.51270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.621 - 1.27i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.21iT - 11T^{2} \) |
| 13 | \( 1 + 5.08iT - 13T^{2} \) |
| 17 | \( 1 + 0.616T + 17T^{2} \) |
| 19 | \( 1 - 4.48iT - 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 + 5.67iT - 29T^{2} \) |
| 31 | \( 1 - 6.91T + 31T^{2} \) |
| 37 | \( 1 - 6.91iT - 37T^{2} \) |
| 41 | \( 1 - 0.616T + 41T^{2} \) |
| 43 | \( 1 - 7.99iT - 43T^{2} \) |
| 47 | \( 1 + 4.97T + 47T^{2} \) |
| 53 | \( 1 - 4.48iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 9.56iT - 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 5.23T + 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−11.39873316371642237501285584837, −10.27568631411612037310694132671, −9.765860166808184914598897563275, −8.051061963274187492921425888447, −7.76870776500293467006418625528, −6.65312077090968522550278510762, −6.02395745706690985302037867393, −4.80161758993674977462207546031, −3.59059082676602374985579561688, −2.58118396245048387723952940934,
0.820140150956060062796269478836, 2.08762099808199299939956451658, 3.78375436680447551412573481508, 4.71117536544338130968831507296, 5.35386426367311937921771459798, 6.57444226657235613926318276715, 8.257581945502088935952201976037, 8.947481214843817276315468589444, 9.438268541877214491347410474816, 10.75235833946735871311148511221