| L(s) = 1 | + (−1.22 − 1.22i)3-s + 2.44·7-s + 2.99i·9-s + 4.89·11-s + 2i·13-s − 6·17-s + 4.89i·19-s + (−2.99 − 2.99i)21-s + 2.44i·23-s + (3.67 − 3.67i)27-s + 9.79i·31-s + (−5.99 − 5.99i)33-s + 2i·37-s + (2.44 − 2.44i)39-s + 6i·41-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)3-s + 0.925·7-s + 0.999i·9-s + 1.47·11-s + 0.554i·13-s − 1.45·17-s + 1.12i·19-s + (−0.654 − 0.654i)21-s + 0.510i·23-s + (0.707 − 0.707i)27-s + 1.75i·31-s + (−1.04 − 1.04i)33-s + 0.328i·37-s + (0.392 − 0.392i)39-s + 0.937i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.407438935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.407438935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4.89iT - 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 12.2iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 7.34T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 4.89iT - 79T^{2} \) |
| 83 | \( 1 + 7.34iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.860084243606516855570359565047, −8.737672774021413456504127094885, −8.249042886553106458055172326098, −6.98681440439635580311093281790, −6.69401745349335308892127954684, −5.62172684328578717489952923136, −4.71745410487922287316781359951, −3.83138794626566215938153338500, −2.05388921748437615148930135847, −1.28989475626296779572636178527,
0.76161168520116107661676695950, 2.34337653681297377569683665906, 3.93198195935572767804944355242, 4.45400378971733279803856660160, 5.37664337051034007770040203822, 6.35092770901810737144985724633, 7.00939425463627261720568104301, 8.234427433013943760397266017626, 9.089522768885479772020604459487, 9.573901740029717374529223769679
