L(s) = 1 | + (0.458 − 0.888i)2-s + (0.580 − 0.814i)3-s + (−0.580 − 0.814i)4-s + (0.989 − 0.142i)5-s + (−0.458 − 0.888i)6-s + (0.371 − 0.928i)7-s + (−0.989 + 0.142i)8-s + (−0.327 − 0.945i)9-s + (0.327 − 0.945i)10-s + (−0.371 − 0.928i)11-s − 12-s + (−0.654 − 0.755i)14-s + (0.458 − 0.888i)15-s + (−0.327 + 0.945i)16-s + (0.888 − 0.458i)17-s + (−0.989 − 0.142i)18-s + ⋯ |
L(s) = 1 | + (0.458 − 0.888i)2-s + (0.580 − 0.814i)3-s + (−0.580 − 0.814i)4-s + (0.989 − 0.142i)5-s + (−0.458 − 0.888i)6-s + (0.371 − 0.928i)7-s + (−0.989 + 0.142i)8-s + (−0.327 − 0.945i)9-s + (0.327 − 0.945i)10-s + (−0.371 − 0.928i)11-s − 12-s + (−0.654 − 0.755i)14-s + (0.458 − 0.888i)15-s + (−0.327 + 0.945i)16-s + (0.888 − 0.458i)17-s + (−0.989 − 0.142i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-1.932071986 - 3.201366534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-1.932071986 - 3.201366534i\) |
\(L(1)\) |
\(\approx\) |
\(0.7502683653 - 1.644935455i\) |
\(L(1)\) |
\(\approx\) |
\(0.7502683653 - 1.644935455i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.458 - 0.888i)T \) |
| 3 | \( 1 + (0.580 - 0.814i)T \) |
| 5 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (0.371 - 0.928i)T \) |
| 11 | \( 1 + (-0.371 - 0.928i)T \) |
| 17 | \( 1 + (0.888 - 0.458i)T \) |
| 19 | \( 1 + (0.945 - 0.327i)T \) |
| 23 | \( 1 + (-0.981 - 0.189i)T \) |
| 29 | \( 1 + (-0.786 + 0.618i)T \) |
| 31 | \( 1 + (0.755 - 0.654i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.0950 - 0.995i)T \) |
| 43 | \( 1 + (0.786 + 0.618i)T \) |
| 47 | \( 1 + (-0.909 - 0.415i)T \) |
| 53 | \( 1 + (0.415 + 0.909i)T \) |
| 59 | \( 1 + (0.814 - 0.580i)T \) |
| 61 | \( 1 + (0.723 - 0.690i)T \) |
| 67 | \( 1 + (0.814 + 0.580i)T \) |
| 71 | \( 1 + (0.618 - 0.786i)T \) |
| 73 | \( 1 + (-0.755 + 0.654i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.540 + 0.841i)T \) |
| 97 | \( 1 + (-0.371 + 0.928i)T \) |
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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
−21.54217500760355396173249041844, −21.07809470858911842613294826094, −20.52262024967838205392788532941, −19.19661939720565907435810614935, −18.211723366948401728218077290389, −17.70265627659728235739172439885, −16.82898264662896693974192091384, −15.95631782810852407086004790902, −15.37539633122336266949867935799, −14.50216906745604348887134239565, −14.23491792956617450914223365294, −13.24875985116825314486983866126, −12.43656327595509753801010355059, −11.516571586102289797343545522157, −10.016737158965974186341176170838, −9.78401590212184882127987819415, −8.77120702460620074797318176680, −8.06608297192508421952688452640, −7.2153951193775365735559435326, −5.898414490833942203838031999203, −5.4455420411752522620881840192, −4.69486292064131427194959449191, −3.60463040871068960543500531244, −2.67937051234172961987243320103, −1.80713908032638049465866144119,
0.5709205476762292445843585473, 1.20333125517079258513457786001, 2.13421361618649179845047229279, 3.04080700841159233256228705364, 3.79881346761861756620994760572, 5.115762973071448101312246201515, 5.78531658032845282337830245730, 6.76027548312590606491376694335, 7.816235978912686409995691872240, 8.683646707759019220835889650620, 9.6227000818767741918239719006, 10.22807124055019308941866975075, 11.22315539076505697887063307154, 11.990617806289442586927469100762, 12.96670133266364133403487818842, 13.52765157046161043468440878080, 14.12519545720731918729663831704, 14.43648631991148526390602575372, 15.83600077907248431375683953789, 16.97482986669046711578876253645, 17.773747918275655218784785473416, 18.43494034502342952081402443573, 19.0379043822480335626785214736, 20.017393578227973180445369832950, 20.6337982480163934465944644100