| L(s) = 1 | − 4-s + 16-s − 4·19-s − 6·29-s − 2·31-s − 6·41-s − 49-s − 6·59-s − 2·61-s − 64-s + 4·76-s − 16·79-s − 12·89-s − 36·101-s − 4·109-s + 6·116-s − 22·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 0.917·19-s − 1.11·29-s − 0.359·31-s − 0.937·41-s − 1/7·49-s − 0.781·59-s − 0.256·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s − 1.27·89-s − 3.58·101-s − 0.383·109-s + 0.557·116-s − 2·121-s + 0.179·124-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 + 0.0783·163-s + 0.468·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9922500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3013376667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3013376667\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.926240425076139529861135002547, −8.365065907294969951319741632313, −8.332642460642684211519171969751, −7.67054796206946527669475760269, −7.53796247359455911496786008721, −6.88069824419769952242560770924, −6.73372141171708547299939532131, −6.20666347786653201093634544482, −5.79772447578655773769943597364, −5.40919411310355706303595223420, −5.09722438654297425079593523972, −4.56497589414483073582353861239, −4.17885976969187413867762469923, −3.79512642451380361793118861554, −3.44089148561824895520842404339, −2.68174709079835084957881115169, −2.49738694601324945227769951780, −1.50070957336960335228174065644, −1.45381212515555590144940713589, −0.17003627585707894682212660472,
0.17003627585707894682212660472, 1.45381212515555590144940713589, 1.50070957336960335228174065644, 2.49738694601324945227769951780, 2.68174709079835084957881115169, 3.44089148561824895520842404339, 3.79512642451380361793118861554, 4.17885976969187413867762469923, 4.56497589414483073582353861239, 5.09722438654297425079593523972, 5.40919411310355706303595223420, 5.79772447578655773769943597364, 6.20666347786653201093634544482, 6.73372141171708547299939532131, 6.88069824419769952242560770924, 7.53796247359455911496786008721, 7.67054796206946527669475760269, 8.332642460642684211519171969751, 8.365065907294969951319741632313, 8.926240425076139529861135002547
