L(s) = 1 | + (−0.898 − 0.438i)5-s + (0.982 + 0.188i)7-s + (−0.988 − 0.150i)11-s + (−0.942 + 0.334i)13-s + (0.489 − 0.872i)17-s + (0.351 + 0.936i)19-s + (−0.553 − 0.832i)23-s + (0.614 + 0.788i)25-s + (0.553 − 0.832i)29-s + (0.999 + 0.0378i)31-s + (−0.800 − 0.599i)35-s + (−0.843 + 0.537i)37-s + (0.997 + 0.0756i)41-s + (0.243 − 0.969i)43-s + (−0.965 − 0.261i)47-s + ⋯ |
L(s) = 1 | + (−0.898 − 0.438i)5-s + (0.982 + 0.188i)7-s + (−0.988 − 0.150i)11-s + (−0.942 + 0.334i)13-s + (0.489 − 0.872i)17-s + (0.351 + 0.936i)19-s + (−0.553 − 0.832i)23-s + (0.614 + 0.788i)25-s + (0.553 − 0.832i)29-s + (0.999 + 0.0378i)31-s + (−0.800 − 0.599i)35-s + (−0.843 + 0.537i)37-s + (0.997 + 0.0756i)41-s + (0.243 − 0.969i)43-s + (−0.965 − 0.261i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.170653847 - 0.7226446448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.170653847 - 0.7226446448i\) |
\(L(1)\) |
\(\approx\) |
\(0.8915484586 - 0.08273921888i\) |
\(L(1)\) |
\(\approx\) |
\(0.8915484586 - 0.08273921888i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 \) |
good | 5 | \( 1 + (-0.898 - 0.438i)T \) |
| 7 | \( 1 + (0.982 + 0.188i)T \) |
| 11 | \( 1 + (-0.988 - 0.150i)T \) |
| 13 | \( 1 + (-0.942 + 0.334i)T \) |
| 17 | \( 1 + (0.489 - 0.872i)T \) |
| 19 | \( 1 + (0.351 + 0.936i)T \) |
| 23 | \( 1 + (-0.553 - 0.832i)T \) |
| 29 | \( 1 + (0.553 - 0.832i)T \) |
| 31 | \( 1 + (0.999 + 0.0378i)T \) |
| 37 | \( 1 + (-0.843 + 0.537i)T \) |
| 41 | \( 1 + (0.997 + 0.0756i)T \) |
| 43 | \( 1 + (0.243 - 0.969i)T \) |
| 47 | \( 1 + (-0.965 - 0.261i)T \) |
| 53 | \( 1 + (-0.726 + 0.686i)T \) |
| 59 | \( 1 + (0.489 + 0.872i)T \) |
| 61 | \( 1 + (-0.132 + 0.991i)T \) |
| 67 | \( 1 + (-0.898 + 0.438i)T \) |
| 71 | \( 1 + (0.0189 - 0.999i)T \) |
| 73 | \( 1 + (0.914 - 0.404i)T \) |
| 79 | \( 1 + (-0.316 - 0.948i)T \) |
| 83 | \( 1 + (0.776 + 0.629i)T \) |
| 89 | \( 1 + (-0.672 - 0.739i)T \) |
| 97 | \( 1 + (-0.999 + 0.0378i)T \) |
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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
−18.30203360832627580741416219344, −17.72419436951317351628851819666, −17.3300708589158747988662593953, −16.19659108061825011579660867017, −15.66286682494257752125170420453, −15.04821238204524041192965191579, −14.427693478164708259350109174031, −13.83638426531235278736960918390, −12.74408974378700904137331234457, −12.32285397000321262999018852178, −11.4062528533775448434973289703, −10.96923383835835634210846387439, −10.25007269140448247778393667426, −9.548459502211855073741851984120, −8.29106663541600032308308009585, −8.01245624921306509652775394707, −7.375232802875519871530538230844, −6.669441860035297595478213398340, −5.49628141114962348420385626467, −4.90954022865592879495353591191, −4.248297023550486793234417126058, −3.26416119763809482337242593761, −2.62202699890613569299848758182, −1.64100296651105566164013517637, −0.58064296206735355708689684536,
0.33262050380743153181251840285, 1.19748613057418173131562526658, 2.28628732380103103968385888888, 2.95583393807097671351141277378, 4.04788511347901187845530385693, 4.75840421944941169052926505923, 5.174019642713017075296790551468, 6.11049114900981763481641651339, 7.27611703693373201320620628140, 7.77953307782119880395742550660, 8.24999945097055375248133594287, 9.02111609945312816876783004599, 10.014991233680336173535788295396, 10.553209369743460615536325585748, 11.53930938289373646959937705759, 12.05538234909761197066265612022, 12.37320765216911304021823557914, 13.53301186196762523453505267529, 14.1275929333992926223930652633, 14.8453246683409578472824814202, 15.488875224138691427110222000726, 16.17993970871597939830145400828, 16.69698540809986003065897024503, 17.55874667098551070731727979526, 18.23745970756702479976291953288