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 | \( 1 + (-0.866 + 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.828056134359104090739729692116, −26.15477484183996775318333625600, −25.31376984072046908682996371519, −24.46833024576671375167671668827, −23.24382574570490539291079758147, −22.688526767996809217396115767789, −21.11869513762016963877915814711, −19.9787418824012243075266056601, −19.4086592732294568144919673273, −18.34826360860243152095053385386, −17.23799974071064912474946976113, −16.63413132467952315955540626230, −15.51939700875446647799235285435, −14.66985834653685647332936093959, −13.44808340131484851702089351555, −12.18754689972420518683824667813, −10.910069240387918540213800516936, −9.8562171150608218372081031862, −9.20808467178953941139441700607, −7.77776832195723862040827138107, −6.913184737502084248287092587799, −5.93793251191954503122222403150, −4.428053053184624712112259218432, −2.73750900513436306530322114607, −1.0418016641120156393284815080,
1.11011687382691575410678988202, 2.81395148206164051328152603066, 3.64962862735336818598197176990, 5.65666621331659245081332515092, 6.82303398218657679367019932937, 8.02510179557561404776012226153, 9.14685144078286352149757633790, 9.83835877194517493476145402531, 11.08920676063131073288119847237, 12.03190508571776441579899504941, 12.91920593388761488619054944430, 14.21020795589167993163483090935, 15.72331345716894296447015615648, 16.42685316750209997153785941381, 17.33128504826037074996413476025, 18.76013073096563086768624748884, 18.96652935557913513609254619493, 20.14666869034656100794412007157, 21.13964420285661760065981122333, 22.03360978180242205963697833728, 22.934057107417441576685929625532, 24.49454979522760405663892506249, 25.271469793929650092873637542, 26.06357209030074734796113752826, 27.17301526627263270510093532279