L(s) = 1 | + (0.854 + 0.854i)3-s + (−3.24 − 3.24i)7-s − 1.53i·9-s + 4.86i·11-s + (4.90 + 4.90i)13-s + (−3.33 − 3.33i)17-s + 1.13·19-s − 5.54i·21-s + (−1.49 − 4.55i)23-s + (3.87 − 3.87i)27-s + 0.233i·29-s + 6.17·31-s + (−4.15 + 4.15i)33-s + (5.16 + 5.16i)37-s + 8.39i·39-s + ⋯ |
L(s) = 1 | + (0.493 + 0.493i)3-s + (−1.22 − 1.22i)7-s − 0.512i·9-s + 1.46i·11-s + (1.36 + 1.36i)13-s + (−0.808 − 0.808i)17-s + 0.260·19-s − 1.20i·21-s + (−0.311 − 0.950i)23-s + (0.746 − 0.746i)27-s + 0.0432i·29-s + 1.10·31-s + (−0.724 + 0.724i)33-s + (0.849 + 0.849i)37-s + 1.34i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.868155180\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.868155180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (1.49 + 4.55i)T \) |
good | 3 | \( 1 + (-0.854 - 0.854i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.24 + 3.24i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.86iT - 11T^{2} \) |
| 13 | \( 1 + (-4.90 - 4.90i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.33 + 3.33i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 29 | \( 1 - 0.233iT - 29T^{2} \) |
| 31 | \( 1 - 6.17T + 31T^{2} \) |
| 37 | \( 1 + (-5.16 - 5.16i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.771T + 41T^{2} \) |
| 43 | \( 1 + (-5.83 + 5.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.739 + 0.739i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.83iT - 59T^{2} \) |
| 61 | \( 1 - 5.08iT - 61T^{2} \) |
| 67 | \( 1 + (-1.97 - 1.97i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + (-1.99 - 1.99i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.05T + 79T^{2} \) |
| 83 | \( 1 + (-7.84 + 7.84i)T - 83iT^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + (10.8 + 10.8i)T + 97iT^{2} \) |
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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
−9.164132708545837152153981204288, −8.426385186202562046477313214648, −7.16123829849751810126677794304, −6.78510433257032654194116944774, −6.17214297168449777875374139363, −4.44994857680220701465160828012, −4.24870891336388744067644915086, −3.38169309691903974419438722536, −2.29329105348739888418053996781, −0.78409442122326209391772983741,
0.966813291718448129939258556196, 2.45195668198308670202767652799, 3.06257356874334974399849982301, 3.83552631686574844651171559639, 5.42916011709207927587032778591, 5.99687862687075992689802512745, 6.41659078512180562040801310595, 7.80725780236081443253988711619, 8.237594743136552058884987464636, 8.905852394857879300209671506006