L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (−0.866 + 0.500i)6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.866 − 1.49i)10-s + (−0.707 − 0.707i)11-s + (−0.965 + 0.258i)12-s + (0.366 − 0.366i)13-s + (−0.258 + 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.707 + 0.707i)3-s + (0.866 + 0.499i)4-s + (−1.22 − 1.22i)5-s + (−0.866 + 0.500i)6-s + i·7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (−0.866 − 1.49i)10-s + (−0.707 − 0.707i)11-s + (−0.965 + 0.258i)12-s + (0.366 − 0.366i)13-s + (−0.258 + 0.965i)14-s + 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.487852121\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487852121\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
good | 5 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 13 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - 1.93T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.737032344243545006942887830057, −7.933928074606190766041037431268, −7.23471009172685199965502751420, −6.10425999936415416901769798122, −5.43940154537612672429560153011, −5.03700578812925240563420007747, −4.33327176608740484043463720650, −3.48459442119935604399606584580, −2.72988614739859624462827227150, −0.821630196187036966879707990220,
1.14608094220888338400354180758, 2.37811722475327698297844180189, 3.37787020493153626111754490817, 3.97742715288329574186804021405, 4.85425537220580614167125952494, 5.68376533320453529160530179190, 6.69087385311231333438315573621, 7.02727857803045116178634325173, 7.62175040108114883151529984919, 8.114151394976741554441188702036