L(s) = 1 | + (0.707 + 0.707i)3-s + (−2.63 + 0.189i)7-s + 1.00i·9-s − 3·11-s + (−1.41 − 1.41i)13-s + (4.24 − 4.24i)17-s + 3.46·19-s + (−2 − 1.73i)21-s + (3.67 − 3.67i)23-s + (−0.707 + 0.707i)27-s + 9i·29-s − 6.92i·31-s + (−2.12 − 2.12i)33-s + (3.67 + 3.67i)37-s − 2.00i·39-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.997 + 0.0716i)7-s + 0.333i·9-s − 0.904·11-s + (−0.392 − 0.392i)13-s + (1.02 − 1.02i)17-s + 0.794·19-s + (−0.436 − 0.377i)21-s + (0.766 − 0.766i)23-s + (−0.136 + 0.136i)27-s + 1.67i·29-s − 1.24i·31-s + (−0.369 − 0.369i)33-s + (0.604 + 0.604i)37-s − 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.636569121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636569121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.189i)T \) |
good | 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (1.41 + 1.41i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + (-3.67 + 3.67i)T - 23iT^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-8.57 + 8.57i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + (-8.57 - 8.57i)T + 67iT^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 + (-1.41 - 1.41i)T + 73iT^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (-2.82 + 2.82i)T - 97iT^{2} \) |
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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
−9.163784699172511157988322428410, −8.390398712241730773569242342817, −7.40306638612315788993879984760, −7.00423997498214721961892466095, −5.60483313288257038851561737626, −5.25670979160756385577609525921, −4.05945129825690415327311256718, −3.01608045353504094043530718889, −2.61536066038738878127445891712, −0.68662947406279345236234715939,
0.991929996041089219402562143838, 2.40430976968436423410155049052, 3.19087276596127390759346006235, 4.04884231994450718252856823438, 5.30434772959619992037723468222, 6.00854872646675687260049608347, 6.88749266923660522325332734927, 7.66917763436091607754684634900, 8.158116970902628391885939476958, 9.355634654506712616039882284433