| L(s) = 1 | − 9·9-s − 52·19-s − 92·31-s + 94·49-s + 148·61-s + 284·79-s + 81·81-s + 428·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 146·169-s + 468·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 9-s − 2.73·19-s − 2.96·31-s + 1.91·49-s + 2.42·61-s + 3.59·79-s + 81-s + 3.92·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.863·169-s + 2.73·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.090465667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090465667\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 94 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 146 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2062 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3214 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 5906 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8542 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 18814 T^{2} + p^{4} T^{4} \) |
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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
−11.65894457312569996036557439796, −11.05554593106336849182783927498, −11.02248816653733503534775540612, −10.44235121939280235955224392967, −9.997613870902732190259176458328, −9.106530733035266101682380655753, −9.017331377980182466971970882402, −8.469124244425378967815587703750, −8.084862373590894508443976560871, −7.32157117934016320868338868758, −6.94767676644916616494534931885, −6.21991501236242035894940905654, −5.88583030440727972690620875082, −5.27202927199347490164345987491, −4.66485083460652070047981802162, −3.80122920765454181374172985339, −3.56643886117874732401939577164, −2.21309151752648367900254771234, −2.18127135462061597737397001789, −0.48535084706982882982774333539,
0.48535084706982882982774333539, 2.18127135462061597737397001789, 2.21309151752648367900254771234, 3.56643886117874732401939577164, 3.80122920765454181374172985339, 4.66485083460652070047981802162, 5.27202927199347490164345987491, 5.88583030440727972690620875082, 6.21991501236242035894940905654, 6.94767676644916616494534931885, 7.32157117934016320868338868758, 8.084862373590894508443976560871, 8.469124244425378967815587703750, 9.017331377980182466971970882402, 9.106530733035266101682380655753, 9.997613870902732190259176458328, 10.44235121939280235955224392967, 11.02248816653733503534775540612, 11.05554593106336849182783927498, 11.65894457312569996036557439796
