| L(s) = 1 | − 3·9-s − 14·13-s + 20·37-s − 13·49-s − 2·61-s − 20·73-s + 9·81-s − 38·97-s − 34·109-s + 42·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s − 3.88·13-s + 3.28·37-s − 1.85·49-s − 0.256·61-s − 2.34·73-s + 81-s − 3.85·97-s − 3.25·109-s + 3.88·117-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 + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4402040681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4402040681\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−9.916155051522485848918594032273, −9.448549833484785043448176587654, −9.422550516650503380920603643725, −8.835027303624975240967525120916, −8.010600507619943124845136573336, −7.953997253661533954297839297987, −7.65386470864920335632622302971, −6.98435933269876262489796397732, −6.83269136474204915474153533449, −6.13952396670763948482909527941, −5.71484093423719012499150660632, −5.15716719821352489369661815286, −4.92828676989730344467296981644, −4.40213043134028745468022780774, −4.01857656322000945710657715467, −2.82239737836733782608150438975, −2.80030907518900402726418495515, −2.47332103456048729065430526665, −1.53537794213936110850949704367, −0.27172348138090443192381610246,
0.27172348138090443192381610246, 1.53537794213936110850949704367, 2.47332103456048729065430526665, 2.80030907518900402726418495515, 2.82239737836733782608150438975, 4.01857656322000945710657715467, 4.40213043134028745468022780774, 4.92828676989730344467296981644, 5.15716719821352489369661815286, 5.71484093423719012499150660632, 6.13952396670763948482909527941, 6.83269136474204915474153533449, 6.98435933269876262489796397732, 7.65386470864920335632622302971, 7.953997253661533954297839297987, 8.010600507619943124845136573336, 8.835027303624975240967525120916, 9.422550516650503380920603643725, 9.448549833484785043448176587654, 9.916155051522485848918594032273
