L(s) = 1 | + (−0.951 − 0.309i)3-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (0.587 − 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (0.587 + 0.809i)12-s + (1.53 + 0.5i)13-s − 0.999·15-s + (0.309 + 0.951i)16-s + (−0.587 − 1.80i)17-s + (−0.951 − 0.309i)20-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + (−0.587 − 0.809i)27-s + (−0.951 + 0.309i)28-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)3-s + (−0.809 − 0.587i)4-s + (0.951 − 0.309i)5-s + (0.587 − 0.809i)7-s + (0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (0.587 + 0.809i)12-s + (1.53 + 0.5i)13-s − 0.999·15-s + (0.309 + 0.951i)16-s + (−0.587 − 1.80i)17-s + (−0.951 − 0.309i)20-s + (−0.809 + 0.587i)21-s + (0.809 − 0.587i)25-s + (−0.587 − 0.809i)27-s + (−0.951 + 0.309i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8313898734\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8313898734\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−9.810508704263537922974220043649, −9.275520524724062952432154976639, −8.232568851027655860698264147531, −7.11658401316404827741790666736, −6.41699918878489643958244253979, −5.43266462350021614940745833288, −4.84273581506120442008873691130, −4.12747945818839430585349040632, −1.97449069812931206431957072707, −1.00435722343690978763363889405,
1.52610180369650430253024447315, 3.18421409198057726799335671910, 4.11552504550977431232957800260, 5.26866980765039813972711126983, 5.85086908110150789959823048200, 6.42717078079068169548461880359, 7.933250503992542678086076837404, 8.668072883512394298791348418783, 9.199799987743475415271538748591, 10.36884265616715813359550188067