L(s) = 1 | − 2-s − 3-s + 4-s + i·5-s + 6-s − i·7-s − 8-s + 9-s − i·10-s − 11-s − 12-s + i·13-s + i·14-s − i·15-s + 16-s + i·17-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s + i·5-s + 6-s − i·7-s − 8-s + 9-s − i·10-s − 11-s − 12-s + i·13-s + i·14-s − i·15-s + 16-s + i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2195118702 + 0.2709838852i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2195118702 + 0.2709838852i\) |
\(L(1)\) |
\(\approx\) |
\(0.4389419288 + 0.1368795992i\) |
\(L(1)\) |
\(\approx\) |
\(0.4389419288 + 0.1368795992i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + iT \) |
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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
−29.19639614232536785519507590907, −28.64218081126164029918950649374, −27.88600270816115195863810751233, −27.0956741314190079725162405451, −25.58381697602523909314146476636, −24.59782078740071861749578257500, −23.891494176626831388615122415857, −22.40790994942075057956329718740, −21.15469955831986685258645842823, −20.34482598142103982359117776175, −18.822271716536776996172143150111, −18.01228598666779850131918941853, −17.12842678425865073792955559960, −15.90902989444569432032956259251, −15.50231464188291418156173209057, −12.96425365199509138847810494123, −12.14445181807835135593646885275, −11.05776802435381329264328325408, −9.84040681941754960017621727161, −8.70967129756331932873721107279, −7.491983993131464172499974983789, −5.89231965169836041380711111927, −5.022159286447815666668939850, −2.45044595969309393949711640008, −0.54941516288487583237355999153,
1.72473267424681441337384180171, 3.73864225832041735256654048605, 5.80465014053441773584730361982, 6.94474657490022917005543743567, 7.7344714553961445615109965343, 9.7447273521408879814251311476, 10.66717335383177216474695952409, 11.21503131695548155823517201909, 12.68949462074814405605998111028, 14.386373554831793385303144665633, 15.77293485108482160268282807993, 16.69051075582370003037072195006, 17.66911429253218547013940794759, 18.52794653564330460394338017565, 19.43182329883066332122445842427, 20.95180207491081621870788410515, 21.88282647600143460534225258107, 23.44032075178551912016186742934, 23.805180486675836008473493099988, 25.57610105101270424110820374080, 26.50985229028967280972834013191, 27.133945159594020552310740058144, 28.352831959863692199237611531116, 29.33841813397548808193969620874, 29.841459466460202099639741131620