L(s) = 1 | + 3-s − 5-s + 6·7-s − 9-s − 3·11-s + 13-s − 15-s − 8·17-s + 2·19-s + 6·21-s + 7·23-s − 5·25-s − 3·29-s + 10·31-s − 3·33-s − 6·35-s + 6·37-s + 39-s + 8·41-s + 15·43-s + 45-s + 5·47-s + 13·49-s − 8·51-s − 13·53-s + 3·55-s + 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 2.26·7-s − 1/3·9-s − 0.904·11-s + 0.277·13-s − 0.258·15-s − 1.94·17-s + 0.458·19-s + 1.30·21-s + 1.45·23-s − 25-s − 0.557·29-s + 1.79·31-s − 0.522·33-s − 1.01·35-s + 0.986·37-s + 0.160·39-s + 1.24·41-s + 2.28·43-s + 0.149·45-s + 0.729·47-s + 13/7·49-s − 1.12·51-s − 1.78·53-s + 0.404·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.506155261\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.506155261\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 62 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 156 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 36 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05653179784718687030000414234, −10.76049969597573691655238784929, −9.967591247771858143259641828511, −9.392410201670798868356218425549, −8.980582610518813772532807858459, −8.665732770087886597582778248719, −8.034577044975961632511233168401, −7.992761883473664672575546172731, −7.49629067270974422852508811169, −7.14734230998654526660491502729, −6.11417336287529391942224171467, −6.06238245304157854839980263678, −4.96591759499945842784207650948, −4.93087888129573414247699269601, −4.38320622430800111044171559562, −3.91433566893515775066259090814, −2.86011579749282015240533827551, −2.53346238926307565915139486861, −1.85961794065305034181282345647, −0.896188645083860498684360941652,
0.896188645083860498684360941652, 1.85961794065305034181282345647, 2.53346238926307565915139486861, 2.86011579749282015240533827551, 3.91433566893515775066259090814, 4.38320622430800111044171559562, 4.93087888129573414247699269601, 4.96591759499945842784207650948, 6.06238245304157854839980263678, 6.11417336287529391942224171467, 7.14734230998654526660491502729, 7.49629067270974422852508811169, 7.992761883473664672575546172731, 8.034577044975961632511233168401, 8.665732770087886597582778248719, 8.980582610518813772532807858459, 9.392410201670798868356218425549, 9.967591247771858143259641828511, 10.76049969597573691655238784929, 11.05653179784718687030000414234