L(s) = 1 | + (0.900 + 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.222 + 0.974i)17-s − 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.974i)29-s − 31-s + (−0.900 + 0.433i)33-s + (−0.222 + 0.974i)37-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)3-s + (−0.900 − 0.433i)5-s + (0.623 + 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.222 + 0.974i)17-s − 19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (0.222 + 0.974i)27-s + (−0.222 + 0.974i)29-s − 31-s + (−0.900 + 0.433i)33-s + (−0.222 + 0.974i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6144154729 + 1.177719431i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6144154729 + 1.177719431i\) |
\(L(1)\) |
\(\approx\) |
\(1.031322483 + 0.3240418538i\) |
\(L(1)\) |
\(\approx\) |
\(1.031322483 + 0.3240418538i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 + (-0.222 + 0.974i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.222 + 0.974i)T \) |
| 29 | \( 1 + (-0.222 + 0.974i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 43 | \( 1 + (0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.623 + 0.781i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.900 - 0.433i)T \) |
| 61 | \( 1 + (-0.222 + 0.974i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + T \) |
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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
−26.50639435683421774376961140, −25.581659596878490743761508763456, −24.44449680441881608182933998802, −23.70519511601669269771354178044, −22.84934181743733125350009962973, −21.44461241257768593702651820950, −20.62901313381340113505827752145, −19.57865750777564188688624393519, −18.774980458575514620556759401093, −18.25818566030721337444425341910, −16.50315070807506250222352434127, −15.63424812829769896669680384911, −14.66664044095100520176926460372, −13.769745049641137869334078096301, −12.783568866373865031325088692740, −11.58933293695329892895281935887, −10.6378085168274211315880149720, −9.094848881472993537302429558759, −8.28446973441073070888462362690, −7.30141761606664784192742054860, −6.30498499872327843403069802866, −4.40495503606128333141823495519, −3.35463700497202313105460548203, −2.24491678769859681471313375613, −0.39681629444658733127715982693,
1.69126300779358280931016876481, 3.25280942213881446231873178553, 4.16918675575237739661255178735, 5.30410284963324420391789511853, 7.163168584806467657610258405563, 8.14161588579281664157813337510, 8.83982686822641588909749159587, 10.18343737658502638247949796752, 11.06951737042613334679864136384, 12.659893376292038703333493766280, 13.17641920487562592587270196052, 14.751971452538730401483066733550, 15.34881705562707543975816431812, 16.12118177219029959163828953134, 17.38931285460657928389232520194, 18.70959681660523407589141274841, 19.61847262334780736347817124380, 20.36579651244330268514806937540, 21.09904349685361160908144366311, 22.282912334216685226297427010827, 23.45859986110124500330356375752, 24.142492524341437498860653559329, 25.6559024034376969467502096995, 25.76813050693921186067460813862, 27.36547192659872327019559801739