| L(s) = 1 | + 4·2-s + 16·4-s − 72·5-s + 64·8-s − 288·10-s + 414·11-s + 1.05e3·13-s + 256·16-s − 1.84e3·17-s − 236·19-s − 1.15e3·20-s + 1.65e3·22-s − 2.89e3·23-s + 2.05e3·25-s + 4.21e3·26-s + 6.52e3·29-s − 6.20e3·31-s + 1.02e3·32-s − 7.39e3·34-s + 9.65e3·37-s − 944·38-s − 4.60e3·40-s + 8.48e3·41-s − 1.08e4·43-s + 6.62e3·44-s − 1.15e4·46-s + 60·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.28·5-s + 0.353·8-s − 0.910·10-s + 1.03·11-s + 1.72·13-s + 1/4·16-s − 1.55·17-s − 0.149·19-s − 0.643·20-s + 0.729·22-s − 1.14·23-s + 0.658·25-s + 1.22·26-s + 1.44·29-s − 1.15·31-s + 0.176·32-s − 1.09·34-s + 1.15·37-s − 0.106·38-s − 0.455·40-s + 0.788·41-s − 0.891·43-s + 0.515·44-s − 0.807·46-s + 0.00396·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.862431345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.862431345\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 72 T + p^{5} T^{2} \) |
| 11 | \( 1 - 414 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1054 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1848 T + p^{5} T^{2} \) |
| 19 | \( 1 + 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 126 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 6522 T + p^{5} T^{2} \) |
| 31 | \( 1 + 200 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 9650 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8484 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10804 T + p^{5} T^{2} \) |
| 47 | \( 1 - 60 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22506 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28176 T + p^{5} T^{2} \) |
| 61 | \( 1 - 35194 T + p^{5} T^{2} \) |
| 67 | \( 1 + 28216 T + p^{5} T^{2} \) |
| 71 | \( 1 - 6642 T + p^{5} T^{2} \) |
| 73 | \( 1 - 52090 T + p^{5} T^{2} \) |
| 79 | \( 1 - 43340 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 98724 T + p^{5} T^{2} \) |
| 97 | \( 1 - 148954 T + p^{5} T^{2} \) |
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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
−9.197541599801431578607086562873, −8.449099818805596024337656419825, −7.70216294707401847066647387877, −6.54237497389872392042359320464, −6.17274731407244635035012561001, −4.63259678258047987938000022627, −4.00976340110350790116165123351, −3.38830862816923451502475401034, −1.92912811425215298046124779733, −0.68470298395690782713088960151,
0.68470298395690782713088960151, 1.92912811425215298046124779733, 3.38830862816923451502475401034, 4.00976340110350790116165123351, 4.63259678258047987938000022627, 6.17274731407244635035012561001, 6.54237497389872392042359320464, 7.70216294707401847066647387877, 8.449099818805596024337656419825, 9.197541599801431578607086562873
