L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + 31-s + (0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (0.866 − 0.5i)23-s + (−0.866 + 0.5i)28-s + (−0.5 − 0.866i)29-s + 31-s + (0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7380609716 + 0.1060281823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7380609716 + 0.1060281823i\) |
\(L(1)\) |
\(\approx\) |
\(0.7200506066 + 0.1014099191i\) |
\(L(1)\) |
\(\approx\) |
\(0.7200506066 + 0.1014099191i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−27.17301526627263270510093532279, −26.06357209030074734796113752826, −25.271469793929650092873637542, −24.49454979522760405663892506249, −22.934057107417441576685929625532, −22.03360978180242205963697833728, −21.13964420285661760065981122333, −20.14666869034656100794412007157, −18.96652935557913513609254619493, −18.76013073096563086768624748884, −17.33128504826037074996413476025, −16.42685316750209997153785941381, −15.72331345716894296447015615648, −14.21020795589167993163483090935, −12.91920593388761488619054944430, −12.03190508571776441579899504941, −11.08920676063131073288119847237, −9.83835877194517493476145402531, −9.14685144078286352149757633790, −8.02510179557561404776012226153, −6.82303398218657679367019932937, −5.65666621331659245081332515092, −3.64962862735336818598197176990, −2.81395148206164051328152603066, −1.11011687382691575410678988202,
1.0418016641120156393284815080, 2.73750900513436306530322114607, 4.428053053184624712112259218432, 5.93793251191954503122222403150, 6.913184737502084248287092587799, 7.77776832195723862040827138107, 9.20808467178953941139441700607, 9.8562171150608218372081031862, 10.910069240387918540213800516936, 12.18754689972420518683824667813, 13.44808340131484851702089351555, 14.66985834653685647332936093959, 15.51939700875446647799235285435, 16.63413132467952315955540626230, 17.23799974071064912474946976113, 18.34826360860243152095053385386, 19.4086592732294568144919673273, 19.9787418824012243075266056601, 21.11869513762016963877915814711, 22.688526767996809217396115767789, 23.24382574570490539291079758147, 24.46833024576671375167671668827, 25.31376984072046908682996371519, 26.15477484183996775318333625600, 26.828056134359104090739729692116