| L(s) = 1 | − 2·7-s + 6·9-s − 4·17-s + 16·23-s + 2·25-s − 16·31-s − 12·41-s − 16·47-s + 3·49-s − 12·63-s + 16·71-s + 12·73-s + 16·79-s + 27·81-s + 12·89-s + 28·97-s − 12·113-s + 8·119-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2·9-s − 0.970·17-s + 3.33·23-s + 2/5·25-s − 2.87·31-s − 1.87·41-s − 2.33·47-s + 3/7·49-s − 1.51·63-s + 1.89·71-s + 1.40·73-s + 1.80·79-s + 3·81-s + 1.27·89-s + 2.84·97-s − 1.12·113-s + 0.733·119-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.077933025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.077933025\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−10.55693802604034433048460987565, −9.637362799388285792536137427651, −9.534353863920074951591938950128, −9.251161596018308489714508592453, −8.769216947924663779715726626624, −8.307575897326408052265656012030, −7.55037972790973719329599722494, −7.30343104806616568061480299580, −6.84998691710577821175779098970, −6.62895490476168478466049476046, −6.30463753630230482359099148310, −5.14410782731278676511379225704, −5.05104511984775916086924410195, −4.79330220502070610909203398861, −3.76252744201132026147857553808, −3.62489600639547292707234514649, −3.07547913068284548591010114276, −2.09994838413178607782110618471, −1.63363373935741372991009952438, −0.72093103111476239774625952082,
0.72093103111476239774625952082, 1.63363373935741372991009952438, 2.09994838413178607782110618471, 3.07547913068284548591010114276, 3.62489600639547292707234514649, 3.76252744201132026147857553808, 4.79330220502070610909203398861, 5.05104511984775916086924410195, 5.14410782731278676511379225704, 6.30463753630230482359099148310, 6.62895490476168478466049476046, 6.84998691710577821175779098970, 7.30343104806616568061480299580, 7.55037972790973719329599722494, 8.307575897326408052265656012030, 8.769216947924663779715726626624, 9.251161596018308489714508592453, 9.534353863920074951591938950128, 9.637362799388285792536137427651, 10.55693802604034433048460987565
