L(s) = 1 | + (1.26 + 1.26i)2-s + 2.17i·4-s + (0.309 − 0.951i)5-s + (1.39 + 1.39i)7-s + (−1.48 + 1.48i)8-s − i·9-s + (1.58 − 0.809i)10-s − 1.17·11-s + 3.52i·14-s − 1.55·16-s + (1.26 − 1.26i)18-s + (2.06 + 0.672i)20-s + (−1.48 − 1.48i)22-s + (−0.809 − 0.587i)25-s + (−3.03 + 3.03i)28-s + 0.618i·29-s + ⋯ |
L(s) = 1 | + (1.26 + 1.26i)2-s + 2.17i·4-s + (0.309 − 0.951i)5-s + (1.39 + 1.39i)7-s + (−1.48 + 1.48i)8-s − i·9-s + (1.58 − 0.809i)10-s − 1.17·11-s + 3.52i·14-s − 1.55·16-s + (1.26 − 1.26i)18-s + (2.06 + 0.672i)20-s + (−1.48 − 1.48i)22-s + (−0.809 − 0.587i)25-s + (−3.03 + 3.03i)28-s + 0.618i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.491820361\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.491820361\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 3 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 11 | \( 1 + 1.17T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 0.618iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.90T + T^{2} \) |
| 43 | \( 1 + (-1.39 + 1.39i)T - iT^{2} \) |
| 47 | \( 1 + (1.26 + 1.26i)T + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1.26 - 1.26i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1 + i)T - iT^{2} \) |
| 89 | \( 1 + 1.17iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.016142504069642522258059663859, −8.557279670288952043749991563508, −8.013366851202917642052776495275, −7.10570851276592936763685471691, −6.06650009947125909204070365697, −5.38997178962007995603470554883, −5.11453788666098100197071892525, −4.26721897665098796983986463774, −3.08161237588720439488492947179, −1.90724786213332537769753829899,
1.49397951306707298342375207183, 2.30715847758054973690222186602, 3.17308709664276454036710089712, 4.22735888723288724380869978315, 4.85203195954914067115284641438, 5.47503551803549820358308711611, 6.57316502014356213471440733092, 7.68610582311065670807739122498, 7.990286224800823111931040551975, 9.700975749065559492443819996733