| L(s) = 1 | + 3·5-s − 9-s − 3·13-s + 2·17-s + 4·25-s + 9·29-s + 37-s + 2·41-s − 3·45-s + 5·49-s + 2·53-s − 3·61-s − 9·65-s − 16·73-s + 81-s + 6·85-s + 7·89-s − 8·97-s + 21·101-s − 20·109-s + 14·113-s + 3·117-s − 8·121-s − 3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1/3·9-s − 0.832·13-s + 0.485·17-s + 4/5·25-s + 1.67·29-s + 0.164·37-s + 0.312·41-s − 0.447·45-s + 5/7·49-s + 0.274·53-s − 0.384·61-s − 1.11·65-s − 1.87·73-s + 1/9·81-s + 0.650·85-s + 0.741·89-s − 0.812·97-s + 2.08·101-s − 1.91·109-s + 1.31·113-s + 0.277·117-s − 0.727·121-s − 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.550975682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.550975682\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.135060148405774351983603011515, −7.76385855650429035752813865105, −7.23737689648776554878723391966, −6.78217546490740551744020748867, −6.36889237058539410650540751902, −5.84813382059573926380683648488, −5.58633460264623558516356272778, −5.06365556259169127496750566121, −4.56572058356245275308335234002, −4.11271810228297644322373094905, −3.17306712933460772316836533092, −2.84575335980601477035712336667, −2.25271521067104098312024403646, −1.64803242018498288650803228475, −0.75937015899210290322448803174,
0.75937015899210290322448803174, 1.64803242018498288650803228475, 2.25271521067104098312024403646, 2.84575335980601477035712336667, 3.17306712933460772316836533092, 4.11271810228297644322373094905, 4.56572058356245275308335234002, 5.06365556259169127496750566121, 5.58633460264623558516356272778, 5.84813382059573926380683648488, 6.36889237058539410650540751902, 6.78217546490740551744020748867, 7.23737689648776554878723391966, 7.76385855650429035752813865105, 8.135060148405774351983603011515
