L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + i·8-s + i·9-s + (−0.707 + 0.707i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + 13-s + (−0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + i·8-s + i·9-s + (−0.707 + 0.707i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + 13-s + (−0.707 − 0.707i)14-s + i·15-s + 16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.5960032172 - 0.6430858675i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.5960032172 - 0.6430858675i\) |
\(L(1)\) |
\(\approx\) |
\(0.4020816796 - 0.6381771585i\) |
\(L(1)\) |
\(\approx\) |
\(0.4020816796 - 0.6381771585i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
| 47 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
| 29 | \( 1 + (-0.707 - 0.707i)T \) |
| 31 | \( 1 + (0.707 + 0.707i)T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + (-0.707 + 0.707i)T \) |
| 43 | \( 1 - iT \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + iT \) |
| 61 | \( 1 + (0.707 - 0.707i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.707 - 0.707i)T \) |
| 73 | \( 1 + (-0.707 - 0.707i)T \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.707 + 0.707i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.60414914469051867406513269477, −22.21552170575680315849459406580, −21.37719293445514728339057756747, −20.39701138855840770920158116416, −19.136558519229769227143762489887, −18.37890061944628235972553547332, −17.669091644002232460920636090689, −17.07197781726204353304104692383, −15.96338966700436795868245064905, −15.34392088576912672300143569577, −14.99100197415405702026912435004, −14.106833503070074327198340210885, −12.88206102171995325027990627002, −11.765763637724753793658888791867, −11.28317785831267864568871576252, −10.242909797874968791738708959372, −9.1824579616050164584914412587, −8.5690895761467750052591769868, −7.37267580776225665911016530346, −6.6512121473342045059284251622, −5.781008629056839800521423929388, −4.8377272773937066191748082558, −4.12733605125074982397939336337, −3.15871165281294742882479435897, −1.21597973596880866619780360400,
0.279795632355618693488424861683, 1.10271051938707929271928320125, 1.71844383878501173279357402839, 3.42526265396799955543567489565, 4.21036025809992843438271751390, 5.08287573619291116558008222546, 6.05351427025237373459125068560, 7.33158166778792463748095831044, 8.32204752004640176920458777070, 8.73319477534947819167091715709, 10.258348349592576069456014989818, 11.02157929841683593111470102877, 11.620069779184299330371631242307, 12.22687700161990462546503426791, 13.19207254877098716085848846710, 13.752574716769626673277533918667, 14.692044057823082237288274819760, 16.19021331556825748328085753590, 16.865608379327863668618774343117, 17.48612206585924578525427772567, 18.50263533176501143863037154539, 19.0838789638717632428006464322, 19.824739479103169772563467759, 20.73319456958272510029734830829, 21.19871245096746904828942535080