| L(s) = 1 | − 7-s + 9-s + 12·11-s + 12·23-s − 10·25-s + 12·29-s + 4·37-s + 8·43-s + 49-s − 12·53-s − 63-s − 16·67-s − 12·71-s − 12·77-s + 8·79-s + 81-s + 12·99-s + 12·107-s + 28·109-s − 12·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1/3·9-s + 3.61·11-s + 2.50·23-s − 2·25-s + 2.22·29-s + 0.657·37-s + 1.21·43-s + 1/7·49-s − 1.64·53-s − 0.125·63-s − 1.95·67-s − 1.42·71-s − 1.36·77-s + 0.900·79-s + 1/9·81-s + 1.20·99-s + 1.16·107-s + 2.68·109-s − 1.12·113-s + 7.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.967734177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967734177\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.468053709441953264102830038362, −7.58881932943940700511157248554, −7.38551790922965845580297625528, −6.71059201271486274282279699706, −6.58911559432542935222512700270, −6.06943423181930863194996495192, −5.84091064569873502208467244139, −4.77020152406148631842736257259, −4.46733872516772697970621091271, −4.14682573537835113219312172425, −3.36010318341016445358722759890, −3.21811200375960790502858283729, −2.18760227953726616802940054530, −1.28052617279627540057290389597, −1.05217073555011886108482433001,
1.05217073555011886108482433001, 1.28052617279627540057290389597, 2.18760227953726616802940054530, 3.21811200375960790502858283729, 3.36010318341016445358722759890, 4.14682573537835113219312172425, 4.46733872516772697970621091271, 4.77020152406148631842736257259, 5.84091064569873502208467244139, 6.06943423181930863194996495192, 6.58911559432542935222512700270, 6.71059201271486274282279699706, 7.38551790922965845580297625528, 7.58881932943940700511157248554, 8.468053709441953264102830038362
