L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (0.781 + 0.623i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.222 + 0.974i)11-s + i·12-s + (0.974 + 0.222i)13-s + (0.900 − 0.433i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (−0.623 − 0.781i)19-s + ⋯ |
L(s) = 1 | + (0.433 − 0.900i)2-s + (−0.781 − 0.623i)3-s + (−0.623 − 0.781i)4-s + (−0.900 + 0.433i)6-s + (0.781 + 0.623i)7-s + (−0.974 + 0.222i)8-s + (0.222 + 0.974i)9-s + (−0.222 + 0.974i)11-s + i·12-s + (0.974 + 0.222i)13-s + (0.900 − 0.433i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.974 + 0.222i)18-s + (−0.623 − 0.781i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.436331765 - 0.2466619243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436331765 - 0.2466619243i\) |
\(L(1)\) |
\(\approx\) |
\(0.9571531428 - 0.4127851963i\) |
\(L(1)\) |
\(\approx\) |
\(0.9571531428 - 0.4127851963i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.433 - 0.900i)T \) |
| 3 | \( 1 + (-0.781 - 0.623i)T \) |
| 7 | \( 1 + (0.781 + 0.623i)T \) |
| 11 | \( 1 + (-0.222 + 0.974i)T \) |
| 13 | \( 1 + (0.974 + 0.222i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (-0.623 - 0.781i)T \) |
| 23 | \( 1 + (0.433 + 0.900i)T \) |
| 31 | \( 1 + (-0.900 - 0.433i)T \) |
| 37 | \( 1 + (0.974 - 0.222i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.433 + 0.900i)T \) |
| 47 | \( 1 + (-0.974 - 0.222i)T \) |
| 53 | \( 1 + (-0.433 + 0.900i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + (0.974 - 0.222i)T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.433 + 0.900i)T \) |
| 79 | \( 1 + (0.222 + 0.974i)T \) |
| 83 | \( 1 + (0.781 - 0.623i)T \) |
| 89 | \( 1 + (0.900 + 0.433i)T \) |
| 97 | \( 1 + (-0.781 + 0.623i)T \) |
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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
−27.56934278437468895413111030180, −27.12025287919228981548638058114, −26.20299748016640646656692033424, −24.94341012090952193296278181000, −23.87406022964119423758960754990, −23.23821419307785812460442002751, −22.3537675148237796247525397048, −21.18688268783908170945176882576, −20.6837558627486683698713737847, −18.531944644789016246364268889059, −17.77654718071183864834592159183, −16.591722422272089846715330157448, −16.18218818799329032817722733366, −14.90344728324111045563773858248, −13.99169126332675607375912567438, −12.81840176380246757126321504358, −11.462045899329491531023873212234, −10.60598319504024448399601130170, −9.0285124774165289564135412735, −7.93383165463786966323833345083, −6.55817736717718835890268810527, −5.55968032115690313474447577406, −4.5256559616151690885111776531, −3.4552146603460830305145420170, −0.60759396806264106672812551665,
1.32016383718193876466957218190, 2.28708691371923823913239326525, 4.23865977603775837555815256163, 5.31123264938614570690159694612, 6.340938206576841171210448616286, 7.96467932861972854415714227951, 9.327969382460257653946455976123, 10.865529855550630090841492764966, 11.351053412008296410005979548103, 12.56730456320636413475233925418, 13.1672652835205628994623355302, 14.55384868013145918367855603906, 15.57641481354792873359743091181, 17.33393472925094270730068943999, 18.04402880925672950875542053671, 18.88973131631103318255568011405, 19.961390524316673688766486142234, 21.24749685122118925400377912277, 21.84593435898391543469220622075, 23.15055934504712047915565755634, 23.621025521776764027039671814313, 24.65505312413982198617858359161, 25.922763366760875507346992796591, 27.74657177383050343801824809496, 27.98613606550198907173798548768