L(s) = 1 | − 2-s + (−0.707 − 0.707i)3-s + 4-s + (1.63 − 1.52i)5-s + (0.707 + 0.707i)6-s + (1.84 + 1.84i)7-s − 8-s + 1.00i·9-s + (−1.63 + 1.52i)10-s − 3.01i·11-s + (−0.707 − 0.707i)12-s − 1.82·13-s + (−1.84 − 1.84i)14-s + (−2.23 − 0.0778i)15-s + 16-s − 5.42i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.731 − 0.682i)5-s + (0.288 + 0.288i)6-s + (0.696 + 0.696i)7-s − 0.353·8-s + 0.333i·9-s + (−0.517 + 0.482i)10-s − 0.907i·11-s + (−0.204 − 0.204i)12-s − 0.504·13-s + (−0.492 − 0.492i)14-s + (−0.577 − 0.0201i)15-s + 0.250·16-s − 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0337 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0337 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.131828794\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.131828794\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.63 + 1.52i)T \) |
| 37 | \( 1 + (-2.27 - 5.64i)T \) |
good | 7 | \( 1 + (-1.84 - 1.84i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.01iT - 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 5.42iT - 17T^{2} \) |
| 19 | \( 1 + (-0.509 + 0.509i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.0216T + 23T^{2} \) |
| 29 | \( 1 + (-4.67 - 4.67i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.55 + 1.55i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.35iT - 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + (0.176 + 0.176i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.05 + 2.05i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.84 + 2.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.51 + 4.51i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.93 - 3.93i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.00T + 71T^{2} \) |
| 73 | \( 1 + (8.50 + 8.50i)T + 73iT^{2} \) |
| 79 | \( 1 + (-2.16 + 2.16i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.925 + 0.925i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.03 + 1.03i)T + 89iT^{2} \) |
| 97 | \( 1 + 7.65iT - 97T^{2} \) |
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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
−9.576813782368923865082931552406, −8.704758953784434850758703748451, −8.269473989619263425740413665962, −7.18190150735109646204849094798, −6.30487987476584033284403260407, −5.37135919818157553548468586695, −4.85696547041282814705718017420, −2.91768235320145381639166727808, −1.89815003938827984612924755446, −0.70614126998679070078507854497,
1.40408887349265738513629898455, 2.50185121800187366452543953949, 3.93083489980825639193090099544, 4.90611985865544569269026186168, 5.99048190649025927696165741460, 6.76551530391997523333826804416, 7.55963561534454658600691500499, 8.388825665576650101026446967273, 9.541156781664266057162596119327, 10.10843180964283949726515689328