| L(s) = 1 | + 2·13-s − 50·25-s + 94·37-s + 71·49-s − 242·61-s + 194·73-s − 334·97-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 − 335·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2/13·13-s − 2·25-s + 2.54·37-s + 1.44·49-s − 3.96·61-s + 2.65·73-s − 3.44·97-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 − 1.98·169-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 + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.869802245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869802245\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 121 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 109 T + p^{2} T^{2} )( 1 + 109 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 131 T + p^{2} T^{2} )( 1 + 131 T + p^{2} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 167 T + p^{2} T^{2} )^{2} \) |
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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
−11.22350048915627375429194764403, −10.82845030045280751140183956497, −10.18700758890995248353834942725, −9.781884316523929509318682268162, −9.416034252577668504245135151005, −9.009090217103188002287093623148, −8.378356030120460568973242699424, −7.83386464557825243703616482515, −7.64520936602000980962755214785, −7.07469443526720345232926839990, −6.28357217520570014258718445151, −6.02525564979307078712264849869, −5.59673200333298115394442559464, −4.78300665068700328445549401352, −4.29787604834519930846402367193, −3.80362616614291217135571169597, −3.04928378165845838475396325124, −2.37095296808941531375312645283, −1.62624688387372412138369279213, −0.58703728566641451083912938177,
0.58703728566641451083912938177, 1.62624688387372412138369279213, 2.37095296808941531375312645283, 3.04928378165845838475396325124, 3.80362616614291217135571169597, 4.29787604834519930846402367193, 4.78300665068700328445549401352, 5.59673200333298115394442559464, 6.02525564979307078712264849869, 6.28357217520570014258718445151, 7.07469443526720345232926839990, 7.64520936602000980962755214785, 7.83386464557825243703616482515, 8.378356030120460568973242699424, 9.009090217103188002287093623148, 9.416034252577668504245135151005, 9.781884316523929509318682268162, 10.18700758890995248353834942725, 10.82845030045280751140183956497, 11.22350048915627375429194764403
