| L(s) = 1 | + 6·3-s − 6·7-s + 18·9-s − 24·11-s + 24·13-s − 24·17-s − 36·21-s + 6·23-s + 54·27-s + 16·31-s − 144·33-s + 96·37-s + 144·39-s − 96·41-s − 54·43-s + 54·47-s + 18·49-s − 144·51-s + 24·53-s + 64·61-s − 108·63-s − 6·67-s + 36·69-s + 96·71-s + 24·73-s + 144·77-s + 243·81-s + ⋯ |
| L(s) = 1 | + 2·3-s − 6/7·7-s + 2·9-s − 2.18·11-s + 1.84·13-s − 1.41·17-s − 1.71·21-s + 6/23·23-s + 2·27-s + 0.516·31-s − 4.36·33-s + 2.59·37-s + 3.69·39-s − 2.34·41-s − 1.25·43-s + 1.14·47-s + 0.367·49-s − 2.82·51-s + 0.452·53-s + 1.04·61-s − 1.71·63-s − 0.0895·67-s + 0.521·69-s + 1.35·71-s + 0.328·73-s + 1.87·77-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.676454660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.676454660\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 782 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96 T + 4608 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T + 1458 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T + 1458 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3362 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10882 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 186 T + 17298 T^{2} - 186 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14942 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.05925817285144134888551415893, −10.72458175013660918899007493321, −10.34956344924694387662957205145, −9.762200093610540505237693532766, −9.454610981170812238437921104651, −8.720518630164331904751601750397, −8.634734448814526651943708523415, −8.046098344605982634052705512875, −7.997532272068241714216158431501, −7.15701527832404135082446591706, −6.62232716123270145599430820839, −6.24667956975343857049687817858, −5.47141942814565829961468910398, −4.81405399984980270801786351033, −4.19306810438204231847214840536, −3.37308028498856193386153598131, −3.22672451318060338373723014491, −2.43870676023969589162237627710, −2.13607627564002681267071726745, −0.72747279136474027612099661197,
0.72747279136474027612099661197, 2.13607627564002681267071726745, 2.43870676023969589162237627710, 3.22672451318060338373723014491, 3.37308028498856193386153598131, 4.19306810438204231847214840536, 4.81405399984980270801786351033, 5.47141942814565829961468910398, 6.24667956975343857049687817858, 6.62232716123270145599430820839, 7.15701527832404135082446591706, 7.997532272068241714216158431501, 8.046098344605982634052705512875, 8.634734448814526651943708523415, 8.720518630164331904751601750397, 9.454610981170812238437921104651, 9.762200093610540505237693532766, 10.34956344924694387662957205145, 10.72458175013660918899007493321, 11.05925817285144134888551415893
