L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯ |
L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 11-s + i·12-s − i·13-s − 14-s + 16-s − i·17-s + i·18-s − 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2508739629 - 0.07126803551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2508739629 - 0.07126803551i\) |
\(L(1)\) |
\(\approx\) |
\(0.3782889217 - 0.6120843330i\) |
\(L(1)\) |
\(\approx\) |
\(0.3782889217 - 0.6120843330i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 61 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + iT \) |
| 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
−25.91550572883776474123639328212, −25.26240111778649551761995997233, −24.01572238674012234469443146773, −23.387540586028269959221253896328, −22.23618112110002296897464178732, −21.60519313944342260774529956592, −20.926260316255417836682305692201, −19.36228888605311807043339997927, −18.56124145054602312755757191976, −17.52202647389983465703245603284, −16.550197945583567777113863301177, −15.89308998663079033462402966222, −15.00834504720961424788336911710, −14.484311484235684443898636789507, −13.18309084125457734335714159350, −12.15189061332745383221591472584, −10.75204219034263757592791511502, −9.85462732364511930394123482419, −8.69891608946843446290951819791, −8.34009808047038041287660347802, −6.666884364065823770027916173419, −5.73086027647352424982592701591, −4.83905060200408314952365652956, −3.88258404783463717893221784523, −2.39648274075954955867456004418,
0.09029431541262276628645005712, 1.02917907859851450752692111560, 2.382834946875695344655796998013, 3.32035203303010820896120832441, 4.75624051294149088786344829000, 5.86290063266501322143183374007, 7.42168216044179468581796958801, 8.01409069890637758854724989871, 9.3021460165744296540779402089, 10.546600296173334770691584380874, 11.115045799098996998468088858191, 12.37245673023316852620387971668, 13.0891000200847953922363387623, 13.70898528592708152786417865098, 14.691919785445769379019552552173, 16.25410741308730948858899166045, 17.64254940953037602370176171889, 17.84192089019378567368929608957, 19.017686036589472872418697425065, 19.78395268868133420677220908510, 20.47402677200106727521195057579, 21.36398132301664815686856507166, 22.74498130299926304328097019873, 23.24207673816330680539021620219, 23.92969305544868970055757352422