L(s) = 1 | + 2·4-s + 3.46i·5-s + 1.73i·7-s + 3.46i·11-s + 4·16-s − 3.46i·19-s + 6.92i·20-s + 6·23-s − 6.99·25-s + 3.46i·28-s − 6·29-s − 1.73i·31-s − 5.99·35-s + 6.92i·41-s − 43-s + 6.92i·44-s + ⋯ |
L(s) = 1 | + 4-s + 1.54i·5-s + 0.654i·7-s + 1.04i·11-s + 16-s − 0.794i·19-s + 1.54i·20-s + 1.25·23-s − 1.39·25-s + 0.654i·28-s − 1.11·29-s − 0.311i·31-s − 1.01·35-s + 1.08i·41-s − 0.152·43-s + 1.04i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.079320168\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.079320168\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 8.66iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 5.19iT - 97T^{2} \) |
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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
−9.840291780894637970062636974721, −9.035578530809179338641130677321, −7.72262467738625320139130400839, −7.22821951012821342910087383198, −6.58144726406100902676289329665, −5.88536566003845186685257809154, −4.74812219646033705570892151709, −3.32399821595725790197619322353, −2.69014699240501934300604652367, −1.85160858216345864639681883169,
0.802733956456812466215252951307, 1.72960496393829983522528904597, 3.19996939244799445412646580023, 4.09028629176335646176633137232, 5.25056178523445820072042224915, 5.80923734757000170175100448326, 6.86764579918031861030213549245, 7.68413198227570546914699010292, 8.430514596693369351002158169863, 9.102409662652452915767317586732