L(s) = 1 | + (0.354 + 0.935i)2-s + (−0.120 + 0.992i)3-s + (−0.748 + 0.663i)4-s + (0.970 + 0.239i)5-s + (−0.970 + 0.239i)6-s + (−0.354 − 0.935i)7-s + (−0.885 − 0.464i)8-s + (−0.970 − 0.239i)9-s + (0.120 + 0.992i)10-s + (0.568 + 0.822i)11-s + (−0.568 − 0.822i)12-s + (−0.748 − 0.663i)13-s + (0.748 − 0.663i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (0.885 − 0.464i)17-s + ⋯ |
L(s) = 1 | + (0.354 + 0.935i)2-s + (−0.120 + 0.992i)3-s + (−0.748 + 0.663i)4-s + (0.970 + 0.239i)5-s + (−0.970 + 0.239i)6-s + (−0.354 − 0.935i)7-s + (−0.885 − 0.464i)8-s + (−0.970 − 0.239i)9-s + (0.120 + 0.992i)10-s + (0.568 + 0.822i)11-s + (−0.568 − 0.822i)12-s + (−0.748 − 0.663i)13-s + (0.748 − 0.663i)14-s + (−0.354 + 0.935i)15-s + (0.120 − 0.992i)16-s + (0.885 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4942964076 + 0.8585990914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4942964076 + 0.8585990914i\) |
\(L(1)\) |
\(\approx\) |
\(0.8133416769 + 0.7752761395i\) |
\(L(1)\) |
\(\approx\) |
\(0.8133416769 + 0.7752761395i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 53 | \( 1 \) |
good | 2 | \( 1 + (0.354 + 0.935i)T \) |
| 3 | \( 1 + (-0.120 + 0.992i)T \) |
| 5 | \( 1 + (0.970 + 0.239i)T \) |
| 7 | \( 1 + (-0.354 - 0.935i)T \) |
| 11 | \( 1 + (0.568 + 0.822i)T \) |
| 13 | \( 1 + (-0.748 - 0.663i)T \) |
| 17 | \( 1 + (0.885 - 0.464i)T \) |
| 19 | \( 1 + (0.748 + 0.663i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.568 - 0.822i)T \) |
| 31 | \( 1 + (-0.568 + 0.822i)T \) |
| 37 | \( 1 + (0.120 - 0.992i)T \) |
| 41 | \( 1 + (-0.568 - 0.822i)T \) |
| 43 | \( 1 + (0.120 + 0.992i)T \) |
| 47 | \( 1 + (-0.970 + 0.239i)T \) |
| 59 | \( 1 + (-0.970 + 0.239i)T \) |
| 61 | \( 1 + (-0.885 - 0.464i)T \) |
| 67 | \( 1 + (0.748 - 0.663i)T \) |
| 71 | \( 1 + (-0.120 - 0.992i)T \) |
| 73 | \( 1 + (-0.885 + 0.464i)T \) |
| 79 | \( 1 + (0.354 - 0.935i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.885 - 0.464i)T \) |
| 97 | \( 1 + (-0.970 - 0.239i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.56261184463008714032850764383, −31.712454315860223450847072097756, −30.51173379889316199027408536748, −29.45778471874089473606031779075, −28.87848810390536285686267240302, −27.83929664416747383175947146360, −25.97968191425698754028868301535, −24.6727320096329733134687688833, −23.875400835595583669746341273461, −22.16985585030938049855571972711, −21.69007142795959756327677567039, −20.04485293876706227893470354243, −18.96351215139006536375939132505, −18.12596653311923533034341657850, −16.80051032042339450821259458419, −14.49579403393510004593269455662, −13.59440310530017895526567896281, −12.44101513902664073837304219754, −11.588134891999928449498507833407, −9.76873689789944420597201463295, −8.653494217623241520566964026462, −6.32592816031808680715932573425, −5.34027812958640334224853391224, −2.90284628429741877190039083080, −1.59086792655426903007158726037,
3.364494201405403378877903230574, 4.82616551079987322049069178809, 6.06994924182853995673295391877, 7.54369421658015629235940997436, 9.5167282833699589851414816214, 10.1445740632411993613359814304, 12.311401609437040617332018215550, 13.9577258005833326767507671423, 14.61519535182024487728919763363, 16.09322380539395313065681109499, 17.06153279856835297012959869397, 17.87843481501607267158272590693, 20.1088044690912189099508509835, 21.30405084722324793578895080028, 22.48025069305149336592311702594, 23.01757576099303267425030499320, 24.843604028920779935072194984567, 25.75146383500437438682557192686, 26.68238894604723656687977019349, 27.6755439078865654590571460940, 29.28663442264638188395562666336, 30.46644680163664179410804505774, 32.09686477223292375670052628643, 32.72068679895308000751795386298, 33.5690466842908764528978597488