L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.900 + 0.433i)5-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.623 − 0.781i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + 17-s + (−0.623 − 0.781i)19-s + (0.222 + 0.974i)20-s + (0.623 − 0.781i)22-s + (0.900 − 0.433i)23-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.900 + 0.433i)5-s + (0.623 − 0.781i)7-s + (−0.222 − 0.974i)8-s + (−0.623 − 0.781i)10-s + (−0.222 + 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + 17-s + (−0.623 − 0.781i)19-s + (0.222 + 0.974i)20-s + (0.623 − 0.781i)22-s + (0.900 − 0.433i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381967654 + 0.06338362945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381967654 + 0.06338362945i\) |
\(L(1)\) |
\(\approx\) |
\(0.9516766977 - 0.04314270546i\) |
\(L(1)\) |
\(\approx\) |
\(0.9516766977 - 0.04314270546i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 5 | \( 1 + (0.900 + 0.433i)T \) |
| 7 | \( 1 + (0.623 - 0.781i)T \) |
| 11 | \( 1 + (-0.222 + 0.974i)T \) |
| 13 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.623 - 0.781i)T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 31 | \( 1 + (0.900 + 0.433i)T \) |
| 37 | \( 1 + (0.222 + 0.974i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.900 - 0.433i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.900 + 0.433i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.623 + 0.781i)T \) |
| 67 | \( 1 + (-0.222 - 0.974i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.900 - 0.433i)T \) |
| 79 | \( 1 + (0.222 + 0.974i)T \) |
| 83 | \( 1 + (-0.623 - 0.781i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (-0.623 - 0.781i)T \) |
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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
−29.89589946739803501439302954641, −29.22398847987132682715029259893, −28.00967049031699070646403851865, −27.38109988476432575669036917173, −26.05001914186383666965591255409, −24.92252852225493202415397396881, −24.61191577411862892109831718138, −23.16181947138156159859449809486, −21.426097988289480280623327948909, −20.80929928233125871292546302158, −19.27884084835661047533018562161, −18.30648688046024662462300907219, −17.36778646807360011377720163223, −16.42832111316412115808242348913, −15.14230741421501729979347680591, −14.1032550030225855659362540029, −12.54293014860853623615879166214, −11.05587089209457491380485186994, −9.91406544201559092567263713727, −8.7454463331391463698612262088, −7.85740449260935651522557533537, −5.966206671137289871073379983422, −5.34763413824134115769102913493, −2.565678035753751254570614190897, −1.03919919180631091417907524243,
1.34387434800209980896321237678, 2.63844840022674518823456759839, 4.55043396554692288353333910642, 6.62157771820450102087838602181, 7.57283617801599922167835136104, 9.14245120538769304615165158096, 10.16964635649284252122431691745, 11.04840894672485534753698764763, 12.45059783154426658717023058653, 13.81254191723627236066348289225, 15.04261261199718345646323284611, 16.778828668132954310437895031367, 17.40596155558616381837933303088, 18.40488681376623509024816076256, 19.524434948673595744753411483861, 20.86839856609466836804934408248, 21.333453857157457433613679271157, 22.824344744603054851120563695001, 24.24402833593110339256779142836, 25.5356715651914600464753490073, 26.17993095429844728574745818395, 27.21846788406926130503325077831, 28.351286611600704243432724474385, 29.27477651256111997424522265857, 30.21748263279678634787048497774