| L(s) = 1 | − 7-s − 9-s + 8·23-s + 6·25-s + 12·29-s + 12·37-s + 49-s − 4·53-s + 63-s − 16·67-s − 16·71-s + 81-s + 16·107-s − 4·109-s − 4·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8·161-s + 163-s + 167-s − 10·169-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1/3·9-s + 1.66·23-s + 6/5·25-s + 2.22·29-s + 1.97·37-s + 1/7·49-s − 0.549·53-s + 0.125·63-s − 1.95·67-s − 1.89·71-s + 1/9·81-s + 1.54·107-s − 0.383·109-s − 0.376·113-s − 2·121-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 + 0.0798·157-s − 0.630·161-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-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
| \(L(1)\) |
\(\approx\) |
\(1.996498812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.996498812\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
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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
−8.283747832140190637614788902310, −7.84647984026607077958510127851, −7.28202483865616118575173689659, −6.94647588085208791055583599143, −6.42626760673661572297909733942, −6.10963948705851002087559103367, −5.58975139776823907333390427700, −4.88727930468054230680663201082, −4.63943667874035208129541153397, −4.17391298292493881277936245508, −3.19540708993635560306556949635, −2.95873888853187175517589045991, −2.54711717557046456991495678410, −1.40308456493577763401270118037, −0.74500009966359147305938262359,
0.74500009966359147305938262359, 1.40308456493577763401270118037, 2.54711717557046456991495678410, 2.95873888853187175517589045991, 3.19540708993635560306556949635, 4.17391298292493881277936245508, 4.63943667874035208129541153397, 4.88727930468054230680663201082, 5.58975139776823907333390427700, 6.10963948705851002087559103367, 6.42626760673661572297909733942, 6.94647588085208791055583599143, 7.28202483865616118575173689659, 7.84647984026607077958510127851, 8.283747832140190637614788902310
