L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s − i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + i·22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s − i·13-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 − 0.866i)19-s + i·22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0333 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0333 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3940914981 - 0.4074788136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3940914981 - 0.4074788136i\) |
\(L(1)\) |
\(\approx\) |
\(0.5259784365 - 0.1599414237i\) |
\(L(1)\) |
\(\approx\) |
\(0.5259784365 - 0.1599414237i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 - iT \) |
| 17 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.866 + 0.5i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + iT \) |
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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
−35.928886150869338157071003557969, −34.81779179651389218949803737421, −33.84810407164582833472657257167, −32.9734071491104142330023578203, −31.03724032898790829925069450985, −29.48835474049905042769289033941, −28.58087648188883751316775995386, −27.636606319691167828325626589449, −26.20805278624132193919093484434, −24.95152094846234389502100099983, −23.79069625323264199036525808920, −22.84792428868406748886644247061, −20.91736057201061889718337743494, −19.189740522871831255464499885724, −18.28167289380249991367582095931, −17.08237789926625500570762941508, −16.10979359357073578779150555585, −14.46249130538229529448216438267, −12.5061863171024189798533139335, −11.06255783167064549152496437350, −9.74251264738153161888673278144, −7.836761687865052724542143333437, −6.651427155432718658422555601447, −5.17305827598066184261186499461, −1.614142252577517778369903397108,
0.58660318278782783810342301815, 3.300405893387460143152213144727, 5.51761052610238754405018686488, 7.45115921623780644906790709824, 9.23858059425651168162544138315, 10.56092859611644771217775593819, 11.53325282543335546082723962884, 12.995570693817099458925329474223, 15.47776606501942698877079110925, 16.56947069915554402805379144156, 17.71511232749917223259956547541, 18.83589264824499655215871770103, 20.48659491098409147068331501989, 21.52356671931436109022274071222, 22.732413352769022420498266007834, 24.3786396986232881309157891134, 25.99300691800533879507516452989, 27.15748830477500801629446399680, 27.98034907519104417487522402386, 29.19033676384501073990784057037, 29.99110452644838179159717983775, 31.77444632416938725784851753182, 33.26638670852837120971248172461, 34.56691511976696396446704353912, 35.11772007639093372883908462483