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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.735525730 - 0.9530837438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.735525730 - 0.9530837438i\) |
\(L(1)\) |
\(\approx\) |
\(1.620050086 - 0.5882352814i\) |
\(L(1)\) |
\(\approx\) |
\(1.620050086 - 0.5882352814i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−27.02412027370164647525635252885, −25.9187409216765427597603920433, −25.24830535005395390282864745358, −24.05982937676700176674865587808, −23.48538697126580967172164620104, −22.615716574521265493855748807731, −21.405269748069399994631275025278, −20.76461178745947382116878015020, −19.84612990029558441990130664439, −18.21863551675562495381459866131, −17.350556636977567055265043522332, −16.44954144300579526881042076374, −15.23856537949536327635447108992, −14.58485730445865791390385587933, −13.52724468800840131938709025752, −12.61428367460888588426133325873, −11.50460433917584366489389245734, −10.53990883848777474286520808700, −8.91662594064037101680902314062, −7.59387301475659339787111976909, −7.035837712678214350069147160062, −5.38268237194734773501175386890, −4.699594662246284629569045990574, −3.382822225506633739312129667852, −1.918755483050532548837690289228,
1.469804954360373472381151485120, 2.76249746267735420758618157493, 4.010712955643669202729737974467, 5.30075460252508615551164032449, 6.02167074678789012462387609508, 7.62865328891577285366623647135, 8.82813859241131379367899309235, 10.324149037223248737624161492618, 11.11629542976639580408978319856, 12.12529376959329520695623600925, 13.03781578248390279464843771527, 14.21754616408542764065186109493, 14.87265946713106394492543813952, 15.95106619036529320961619702431, 17.13105866964184176029238920526, 18.685884172808354275244286999698, 19.042546686320868596482942354761, 20.63360663698645416924492085189, 21.08931239304821977636081278600, 21.96510228032533054741410346477, 23.05240132896572037477873306781, 23.95470268263317914466146528857, 24.64171421579552876523877646039, 25.68662025927038289628853346799, 27.15042030010197476961649012655