L(s) = 1 | + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (−0.809 − 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (−0.453 + 0.891i)13-s + (0.707 − 0.707i)14-s + (0.156 − 0.987i)15-s + (0.309 − 0.951i)16-s + (−0.987 + 0.156i)17-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.707 + 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (0.891 + 0.453i)6-s + (0.891 − 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (−0.809 − 0.587i)10-s + (0.156 + 0.987i)11-s + (0.987 + 0.156i)12-s + (−0.453 + 0.891i)13-s + (0.707 − 0.707i)14-s + (0.156 − 0.987i)15-s + (0.309 − 0.951i)16-s + (−0.987 + 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.927373363 - 0.4032445846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.927373363 - 0.4032445846i\) |
\(L(1)\) |
\(\approx\) |
\(2.104027268 - 0.2082876971i\) |
\(L(1)\) |
\(\approx\) |
\(2.104027268 - 0.2082876971i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (0.891 - 0.453i)T \) |
| 11 | \( 1 + (0.156 + 0.987i)T \) |
| 13 | \( 1 + (-0.453 + 0.891i)T \) |
| 17 | \( 1 + (-0.987 + 0.156i)T \) |
| 19 | \( 1 + (-0.453 - 0.891i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (-0.987 - 0.156i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (0.951 - 0.309i)T \) |
| 47 | \( 1 + (0.891 + 0.453i)T \) |
| 53 | \( 1 + (0.987 + 0.156i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.951 - 0.309i)T \) |
| 67 | \( 1 + (-0.156 + 0.987i)T \) |
| 71 | \( 1 + (-0.156 - 0.987i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.891 - 0.453i)T \) |
| 97 | \( 1 + (0.156 - 0.987i)T \) |
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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
−34.66425176352297951288214374810, −33.59913951115444738863967937415, −31.86257315558216568197616877782, −31.32719697976162095543389141098, −30.178349726155191225846251735222, −29.547342888474785027065517147564, −27.26917095488785433653471280437, −26.10737445868220921270251305119, −24.772505498930938967935757148341, −24.103175606848655435122298847862, −22.79477740692756125506512047952, −21.52963480252251286107592737702, −20.19834951926494660400617898427, −18.967481037266530124871238388986, −17.584835359070717908431176866153, −15.52507704188104891139350123603, −14.69280022726624804819448394012, −13.66980510159571693117356199282, −12.18585272548723553661009229181, −11.07945140925830638125061391896, −8.35117930629260651266179586654, −7.40553143818840763390146956124, −5.88447004614549439130345827281, −3.753301945376491067748609998865, −2.36490278380049334141559350629,
2.0596443432589604164575716242, 4.27448147541150950147338892159, 4.6920076043228126341351469018, 7.29436541067047397848501848483, 8.96266504287673162023166603896, 10.62998788634676038963245831605, 11.95340058196998665049844836823, 13.43739443363329421394436981697, 14.683221841526731415370067333659, 15.59846819740329598525132726017, 16.98819538978543739002615856652, 19.44595876408721134865614997742, 20.316424153966181041002026927306, 21.076831387115838512502583874076, 22.3878169997118998776124429520, 23.88568752132014274651849435950, 24.66021924370373337973376271310, 26.2743400841518926730974664583, 27.67985093306591755699899461099, 28.5695043391108741066398760893, 30.49126293052938243079234783759, 31.07849561686181681352470507448, 32.1581399752007298570653522503, 33.1189203353543178192161318018, 34.057406252223451788926743556541