L(s) = 1 | + (−1.59 + 0.180i)2-s + (0.149 + 0.988i)3-s + (1.54 − 0.353i)4-s + (−0.416 − 1.55i)6-s + (−0.895 + 0.313i)8-s + (−0.955 + 0.294i)9-s + (0.580 + 1.47i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (1.47 − 0.643i)18-s + (0.781 − 0.623i)23-s + (−0.443 − 0.838i)24-s + (−0.222 − 0.974i)25-s + (1.69 + 2.69i)26-s + (−0.433 − 0.900i)27-s + ⋯ |
L(s) = 1 | + (−1.59 + 0.180i)2-s + (0.149 + 0.988i)3-s + (1.54 − 0.353i)4-s + (−0.416 − 1.55i)6-s + (−0.895 + 0.313i)8-s + (−0.955 + 0.294i)9-s + (0.580 + 1.47i)12-s + (−0.858 − 1.78i)13-s + (−0.0567 + 0.0273i)16-s + (1.47 − 0.643i)18-s + (0.781 − 0.623i)23-s + (−0.443 − 0.838i)24-s + (−0.222 − 0.974i)25-s + (1.69 + 2.69i)26-s + (−0.433 − 0.900i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3791931340\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3791931340\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.149 - 0.988i)T \) |
| 23 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 + (0.680 + 0.733i)T \) |
good | 2 | \( 1 + (1.59 - 0.180i)T + (0.974 - 0.222i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 13 | \( 1 + (0.858 + 1.78i)T + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 31 | \( 1 + (-0.0579 - 0.514i)T + (-0.974 + 0.222i)T^{2} \) |
| 37 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 41 | \( 1 + (-0.922 + 0.922i)T - iT^{2} \) |
| 43 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 47 | \( 1 + (-0.0246 + 0.0705i)T + (-0.781 - 0.623i)T^{2} \) |
| 53 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + 0.445iT - T^{2} \) |
| 61 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 67 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (1.67 - 0.807i)T + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.197 + 1.75i)T + (-0.974 - 0.222i)T^{2} \) |
| 79 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 83 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (0.433 + 0.900i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.317500993856453718514265040136, −8.555476775006910411435631316620, −7.976315031443506571811032036834, −7.33158945690752095081958659751, −6.22157426623057180155281846220, −5.34395744439267694143868670167, −4.42319794278350964229546064996, −3.13849797584546453276943521911, −2.28446777965556325854627972817, −0.46631641788245242496597764047,
1.34704343865213695524546033135, 2.04995994478406305479024245500, 3.08036491626361064034784785001, 4.55057758438707294230821504035, 5.79301009591320186822018727975, 6.86473025130129406926147897593, 7.21309014014431188038728810704, 7.86720748497686875081525666474, 8.794987040771166178497253950126, 9.312130691303281009317111499364