L(s) = 1 | + (0.959 − 0.281i)3-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯ |
L(s) = 1 | + (0.959 − 0.281i)3-s + (0.841 + 0.540i)5-s + (0.142 + 0.989i)7-s + (0.841 − 0.540i)9-s + (−0.415 + 0.909i)11-s + (−0.142 + 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.654 − 0.755i)17-s + (0.654 − 0.755i)19-s + (0.415 + 0.909i)21-s + (0.415 + 0.909i)25-s + (0.654 − 0.755i)27-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (−0.142 + 0.989i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.523006399 + 0.7966350128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.523006399 + 0.7966350128i\) |
\(L(1)\) |
\(\approx\) |
\(1.664099662 + 0.2521117494i\) |
\(L(1)\) |
\(\approx\) |
\(1.664099662 + 0.2521117494i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.142 + 0.989i)T \) |
| 11 | \( 1 + (-0.415 + 0.909i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.654 - 0.755i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 29 | \( 1 + (-0.654 - 0.755i)T \) |
| 31 | \( 1 + (0.959 + 0.281i)T \) |
| 37 | \( 1 + (0.841 - 0.540i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.142 - 0.989i)T \) |
| 59 | \( 1 + (0.142 - 0.989i)T \) |
| 61 | \( 1 + (-0.959 - 0.281i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (-0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.654 + 0.755i)T \) |
| 79 | \( 1 + (0.142 - 0.989i)T \) |
| 83 | \( 1 + (-0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
| 97 | \( 1 + (0.841 + 0.540i)T \) |
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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
−29.93239438232885163740810799687, −29.19440051361174748545954306948, −27.72396988350447265098785155938, −26.68938852942633904760851517241, −25.913009682354216833916646826, −24.74493282143295341821378972551, −24.04726744485761339423450611624, −22.389925627801686658151261049857, −21.22534851963113908415751161053, −20.48692882765145820113135742720, −19.60780051837873686584273702507, −18.17987389020366398679441067369, −16.96756712718466724494575439007, −15.93906995347050715877957503903, −14.541958912639368466023490877822, −13.57277494497270203483748466697, −12.88541105029446172888816194415, −10.745365566031093916485268046980, −9.90376655919993617854976701064, −8.60994566067675723214570423952, −7.62830084641279140414470106759, −5.817269803417478752286115829095, −4.34908874496239890165935393865, −2.924150089646087674392178613138, −1.23746508226654160280972314067,
1.98042502384307438314399426252, 2.75136560435352112906962809071, 4.74051861251371966470145629458, 6.41370946766264094628624189655, 7.53344283775477472208613367789, 9.12692205686240558372928941263, 9.70107500645393315357999743643, 11.475758308505580219410687963669, 12.85426869861500656411058124790, 13.88557314628971299800971584066, 14.84087730331706567540233617347, 15.80080426788559980902456430469, 17.72556587424257395687483195701, 18.36582283556494309418560589046, 19.42898132289104528389860054543, 20.75891081527963857783071198869, 21.53245277151877106951315835109, 22.65174715332329774873541664593, 24.28476240068823914147401737343, 25.01806784720667927479242910591, 25.983288588464404052300477418634, 26.67012432308277566942206885036, 28.31393177594578368841766404099, 29.16986738220082955205471296950, 30.46753561247343296462977602824