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
−10.10843180964283949726515689328, −9.541156781664266057162596119327, −8.388825665576650101026446967273, −7.55963561534454658600691500499, −6.76551530391997523333826804416, −5.99048190649025927696165741460, −4.90611985865544569269026186168, −3.93083489980825639193090099544, −2.50185121800187366452543953949, −1.40408887349265738513629898455,
0.70614126998679070078507854497, 1.89815003938827984612924755446, 2.91768235320145381639166727808, 4.85696547041282814705718017420, 5.37135919818157553548468586695, 6.30487987476584033284403260407, 7.18190150735109646204849094798, 8.269473989619263425740413665962, 8.704758953784434850758703748451, 9.576813782368923865082931552406