| L(s) = 1 | − 3·3-s + 3·7-s + 6·9-s − 6·19-s − 9·21-s + 8·25-s − 9·27-s − 9·29-s + 6·31-s + 6·37-s − 15·47-s + 2·49-s − 9·53-s + 18·57-s − 12·59-s + 18·63-s − 24·75-s + 9·81-s − 15·83-s + 27·87-s − 18·93-s − 6·103-s − 18·111-s + 27·113-s − 12·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.13·7-s + 2·9-s − 1.37·19-s − 1.96·21-s + 8/5·25-s − 1.73·27-s − 1.67·29-s + 1.07·31-s + 0.986·37-s − 2.18·47-s + 2/7·49-s − 1.23·53-s + 2.38·57-s − 1.56·59-s + 2.26·63-s − 2.77·75-s + 81-s − 1.64·83-s + 2.89·87-s − 1.86·93-s − 0.591·103-s − 1.70·111-s + 2.53·113-s − 1.09·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.432836198463597486149328755620, −8.050160623992353513743363763081, −7.50547118111690756540173460213, −7.04206038819788460169294925765, −6.44171806597024004178258496488, −6.22056409017451747702989808040, −5.69109468279723566186540139034, −5.04004363215558633657932131885, −4.64972987661208186595738004351, −4.50916617892083646666542145946, −3.65686786420722870998736063605, −2.76958460171482420780189928660, −1.79952763418592248967346027622, −1.24416744807409352638624365519, 0,
1.24416744807409352638624365519, 1.79952763418592248967346027622, 2.76958460171482420780189928660, 3.65686786420722870998736063605, 4.50916617892083646666542145946, 4.64972987661208186595738004351, 5.04004363215558633657932131885, 5.69109468279723566186540139034, 6.22056409017451747702989808040, 6.44171806597024004178258496488, 7.04206038819788460169294925765, 7.50547118111690756540173460213, 8.050160623992353513743363763081, 8.432836198463597486149328755620
