L(s) = 1 | + (−0.125 − 0.992i)5-s + (0.248 − 0.968i)9-s + (−0.0319 − 0.0540i)13-s + (−0.967 − 1.42i)17-s + (−0.968 + 0.248i)25-s + (−0.666 − 0.313i)29-s + (−0.775 + 1.79i)37-s + (1.05 − 1.65i)41-s + (−0.992 − 0.125i)45-s + (−0.951 + 0.309i)49-s + (0.0212 + 0.677i)53-s + (−0.134 − 0.211i)61-s + (−0.0496 + 0.0385i)65-s + (0.418 − 0.369i)73-s + (−0.876 − 0.481i)81-s + ⋯ |
L(s) = 1 | + (−0.125 − 0.992i)5-s + (0.248 − 0.968i)9-s + (−0.0319 − 0.0540i)13-s + (−0.967 − 1.42i)17-s + (−0.968 + 0.248i)25-s + (−0.666 − 0.313i)29-s + (−0.775 + 1.79i)37-s + (1.05 − 1.65i)41-s + (−0.992 − 0.125i)45-s + (−0.951 + 0.309i)49-s + (0.0212 + 0.677i)53-s + (−0.134 − 0.211i)61-s + (−0.0496 + 0.0385i)65-s + (0.418 − 0.369i)73-s + (−0.876 − 0.481i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9479400844\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9479400844\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.125 + 0.992i)T \) |
good | 3 | \( 1 + (-0.248 + 0.968i)T^{2} \) |
| 7 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 11 | \( 1 + (-0.187 - 0.982i)T^{2} \) |
| 13 | \( 1 + (0.0319 + 0.0540i)T + (-0.481 + 0.876i)T^{2} \) |
| 17 | \( 1 + (0.967 + 1.42i)T + (-0.368 + 0.929i)T^{2} \) |
| 19 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 23 | \( 1 + (-0.904 - 0.425i)T^{2} \) |
| 29 | \( 1 + (0.666 + 0.313i)T + (0.637 + 0.770i)T^{2} \) |
| 31 | \( 1 + (-0.929 - 0.368i)T^{2} \) |
| 37 | \( 1 + (0.775 - 1.79i)T + (-0.684 - 0.728i)T^{2} \) |
| 41 | \( 1 + (-1.05 + 1.65i)T + (-0.425 - 0.904i)T^{2} \) |
| 43 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 47 | \( 1 + (-0.844 - 0.535i)T^{2} \) |
| 53 | \( 1 + (-0.0212 - 0.677i)T + (-0.998 + 0.0627i)T^{2} \) |
| 59 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 61 | \( 1 + (0.134 + 0.211i)T + (-0.425 + 0.904i)T^{2} \) |
| 67 | \( 1 + (0.770 + 0.637i)T^{2} \) |
| 71 | \( 1 + (0.535 - 0.844i)T^{2} \) |
| 73 | \( 1 + (-0.418 + 0.369i)T + (0.125 - 0.992i)T^{2} \) |
| 79 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 83 | \( 1 + (0.248 + 0.968i)T^{2} \) |
| 89 | \( 1 + (-1.53 - 0.0967i)T + (0.992 + 0.125i)T^{2} \) |
| 97 | \( 1 + (0.448 + 1.24i)T + (-0.770 + 0.637i)T^{2} \) |
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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
−8.495013985994579358062365632007, −7.62593617837046680925952886273, −6.93209002914580782799497182355, −6.16868227450346861090002534303, −5.27432537112189194829644125516, −4.59819539387075887918963955801, −3.87222519777569142027687464364, −2.88460325475332306984436479463, −1.69715206069098954973677213301, −0.51056899168246572856474907228,
1.78438947436148931837801963950, 2.44282506920790473116604141680, 3.56082626851589877003263050359, 4.22759537988097114593788442185, 5.17551797974435950828463966968, 6.07852124172147611311538305834, 6.67739642322542187144380032479, 7.49258755898252757625399848547, 8.003468656478580529387136164260, 8.814950528955203079432071443413