L(s) = 1 | + (0.755 + 0.654i)7-s + (0.540 + 0.841i)11-s + (−0.654 − 0.755i)13-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.959 + 0.281i)37-s + (−0.959 − 0.281i)41-s + (0.142 + 0.989i)43-s − i·47-s + (0.142 + 0.989i)49-s + (−0.654 + 0.755i)53-s + (−0.755 + 0.654i)59-s + (−0.989 − 0.142i)61-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)7-s + (0.540 + 0.841i)11-s + (−0.654 − 0.755i)13-s + (−0.909 − 0.415i)17-s + (0.909 − 0.415i)19-s + (0.909 + 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.959 + 0.281i)37-s + (−0.959 − 0.281i)41-s + (0.142 + 0.989i)43-s − i·47-s + (0.142 + 0.989i)49-s + (−0.654 + 0.755i)53-s + (−0.755 + 0.654i)59-s + (−0.989 − 0.142i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7390086477 + 1.112551415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7390086477 + 1.112551415i\) |
\(L(1)\) |
\(\approx\) |
\(1.038772468 + 0.2075236530i\) |
\(L(1)\) |
\(\approx\) |
\(1.038772468 + 0.2075236530i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + (0.755 + 0.654i)T \) |
| 11 | \( 1 + (0.540 + 0.841i)T \) |
| 13 | \( 1 + (-0.654 - 0.755i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (0.909 - 0.415i)T \) |
| 29 | \( 1 + (0.909 + 0.415i)T \) |
| 31 | \( 1 + (-0.142 + 0.989i)T \) |
| 37 | \( 1 + (-0.959 + 0.281i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.654 + 0.755i)T \) |
| 59 | \( 1 + (-0.755 + 0.654i)T \) |
| 61 | \( 1 + (-0.989 - 0.142i)T \) |
| 67 | \( 1 + (-0.841 - 0.540i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (-0.909 + 0.415i)T \) |
| 79 | \( 1 + (0.654 + 0.755i)T \) |
| 83 | \( 1 + (0.959 - 0.281i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.281 - 0.959i)T \) |
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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
−17.417595719000498728704478723905, −17.17411694642627348649343405908, −16.39832326666628953815897379410, −15.76421564020527100374822053057, −14.89993718100923711597462257548, −14.28416679282840981258898146715, −13.76632995040961774547974387092, −13.26003769567941097112686568954, −12.04283570207044166354405914575, −11.80093486964446790907139600206, −10.957075155833292332719523009989, −10.45655342913238525921485238616, −9.530020678565891812358175500796, −8.94907141906045131486106385898, −8.14381659291697553457771472454, −7.57131561083822989347010441885, −6.746071247733847788865785698919, −6.1741323327168097736826818984, −5.20483117303582344789750503979, −4.53094477773578134899615939806, −3.890123260963973179996910065081, −3.10669269382989037687249958255, −2.0154509875578374517753480373, −1.43400989867673380506071165635, −0.34265095765640076105042302364,
1.12387272068110665629940066314, 1.87879066961189213193707674083, 2.69948518597421048284681651503, 3.36741171582011306470591471400, 4.703819058241769490038537731494, 4.80106908406284410646292425354, 5.648602551058641306861513509989, 6.65324302582998284737837349608, 7.19981422954510186996300891385, 7.93922153809757172152211505765, 8.74646540147036895247052780397, 9.25028703569500182367272012269, 10.05260299527015301563689478882, 10.74425844407921610323202248338, 11.520488251942884318242834968606, 12.20011206418230749081620741445, 12.489197236748163330844081972330, 13.61692274951208006950931216954, 14.08165286770915762493918584215, 14.93984097736330474608516299662, 15.339656501502600364369878304878, 15.933096582690889357607464285456, 16.88731772792616370961776200763, 17.682292212875109455023386535365, 17.85922007472880390011201325671