L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.912 − 1.47i)3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.396 + 1.68i)6-s − 2.79·7-s + (0.707 − 0.707i)8-s + (−1.33 + 2.68i)9-s − 1.00·10-s + 4.26·11-s + (1.47 − 0.912i)12-s + (1.77 + 1.77i)13-s + (1.97 + 1.97i)14-s + (−1.68 − 0.396i)15-s − 1.00·16-s + (5.48 − 5.48i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.526 − 0.850i)3-s + 0.500i·4-s + (0.316 − 0.316i)5-s + (−0.161 + 0.688i)6-s − 1.05·7-s + (0.250 − 0.250i)8-s + (−0.445 + 0.895i)9-s − 0.316·10-s + 1.28·11-s + (0.425 − 0.263i)12-s + (0.492 + 0.492i)13-s + (0.528 + 0.528i)14-s + (−0.435 − 0.102i)15-s − 0.250·16-s + (1.33 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9247558138\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9247558138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.912 + 1.47i)T \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-2.97 + 5.30i)T \) |
good | 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.48 + 5.48i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.485 - 0.485i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.36 + 1.36i)T - 23iT^{2} \) |
| 29 | \( 1 + (7.02 + 7.02i)T + 29iT^{2} \) |
| 31 | \( 1 + (3.79 - 3.79i)T - 31iT^{2} \) |
| 41 | \( 1 - 7.88T + 41T^{2} \) |
| 43 | \( 1 + (3.72 + 3.72i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.41iT - 47T^{2} \) |
| 53 | \( 1 + 2.32iT - 53T^{2} \) |
| 59 | \( 1 + (4.84 - 4.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (-3.08 + 3.08i)T - 61iT^{2} \) |
| 67 | \( 1 - 0.631iT - 67T^{2} \) |
| 71 | \( 1 - 12.9iT - 71T^{2} \) |
| 73 | \( 1 + 8.24iT - 73T^{2} \) |
| 79 | \( 1 + (4.17 + 4.17i)T + 79iT^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 + (1.13 + 1.13i)T + 89iT^{2} \) |
| 97 | \( 1 + (-3.82 - 3.82i)T + 97iT^{2} \) |
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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
−9.378906773646668350100669029247, −9.056332854761954311738336437960, −7.77631638159518893091066643833, −7.06507234402808226494066061238, −6.25876551761308413132315921244, −5.48792269165637597122823329404, −4.07207084497845777297393402782, −2.97921451820358750031084051309, −1.69857365180586870531777654281, −0.60418440388070672202007847282,
1.24998534171413282447470485933, 3.29462551055338008947298107602, 3.87880419591629040923486820148, 5.34833675322472207785048057251, 6.07385310559841402151138483607, 6.53650979551321435901695440371, 7.63579554490787924681293248016, 8.793358558785860870847700623629, 9.468449836141004642000453913730, 9.941783443039308481842425808198