L(s) = 1 | + (−0.978 + 0.207i)3-s + (−0.913 − 0.406i)5-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)15-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s − 23-s + (0.669 + 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.913 − 0.406i)31-s + (0.309 + 0.951i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s − 45-s + ⋯ |
L(s) = 1 | + (−0.978 + 0.207i)3-s + (−0.913 − 0.406i)5-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)15-s + (0.809 − 0.587i)17-s + (0.669 + 0.743i)19-s − 23-s + (0.669 + 0.743i)25-s + (−0.809 + 0.587i)27-s + (−0.978 − 0.207i)29-s + (0.913 − 0.406i)31-s + (0.309 + 0.951i)37-s + (−0.669 − 0.743i)41-s + (0.5 + 0.866i)43-s − 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5205212692 + 0.4120200095i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5205212692 + 0.4120200095i\) |
\(L(1)\) |
\(\approx\) |
\(0.6544535540 + 0.03540758185i\) |
\(L(1)\) |
\(\approx\) |
\(0.6544535540 + 0.03540758185i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.669 - 0.743i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.978 + 0.207i)T \) |
| 53 | \( 1 + (0.913 - 0.406i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.104 + 0.994i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.913 - 0.406i)T \) |
| 73 | \( 1 + (0.978 + 0.207i)T \) |
| 79 | \( 1 + (0.104 - 0.994i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.104 - 0.994i)T \) |
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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
−18.42428116778805011253646433801, −17.67000832892500315232590442088, −16.98872823682095212164929240477, −16.243900071477769090163514095995, −15.772929909320557873755343068395, −15.07783021451465719102478962915, −14.29170428218499789045378396086, −13.48309296571204871668733821564, −12.60775609310365326547494612852, −12.11194190754329143590626624312, −11.46248231630246287751124982974, −10.92503040911351238618023088760, −10.19170702338223706565123805773, −9.51710231941601892964380345572, −8.33309023425805223841386358760, −7.760461558319587400125352700581, −7.067802910758725679320138551930, −6.43538990480194368803067893093, −5.59677094069190199367649438473, −4.91717336381274044337330989820, −4.0263708554781240920355465440, −3.4278518052689701559053826718, −2.324568664362197952903826729538, −1.29402664979930543132009035092, −0.315936292945654355330112869094,
0.79718706511084750024433363999, 1.55617005307007348090705362296, 2.92929101212122577661526735835, 3.83342629601239765437867686006, 4.34431955219847158162734160177, 5.247792557726297002298690366109, 5.74820993032766880006643177800, 6.659828670796353827031618169806, 7.52071651927321785429364224676, 7.95962766439304816412680459846, 8.937072216848246968170609909374, 9.91419562251900438392114957992, 10.1909637515519275478148676998, 11.45866037161536184321333521956, 11.60750940829103499979935380408, 12.26194109017522242712260823842, 12.960434228368413419715733464131, 13.79367065230247622065024162240, 14.74072746928452386114392212774, 15.380379204317271991840474192502, 16.117646075543421759097424885421, 16.4932663248714005494103567186, 17.07737228434019419344163687194, 18.02323473752065250582774190081, 18.52568833182774026066679310446