| L(s) = 1 | + 11·7-s − 23·13-s + 74·19-s − 25·25-s − 46·31-s − 146·37-s − 22·43-s + 49·49-s − 47·61-s − 13·67-s + 286·73-s + 11·79-s − 253·91-s + 169·97-s − 157·103-s − 428·109-s − 121·121-s + 127-s + 131-s + 814·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 11/7·7-s − 1.76·13-s + 3.89·19-s − 25-s − 1.48·31-s − 3.94·37-s − 0.511·43-s + 49-s − 0.770·61-s − 0.194·67-s + 3.91·73-s + 0.139·79-s − 2.78·91-s + 1.74·97-s − 1.52·103-s − 3.92·109-s − 121-s + 0.00787·127-s + 0.00763·131-s + 6.12·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.300862105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300862105\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 59 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 73 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )( 1 + 83 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 121 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 109 T + p^{2} T^{2} )( 1 + 122 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 131 T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )( 1 - 2 T + p^{2} T^{2} ) \) |
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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
−9.491916162995338632485745474389, −9.348533940456258489932083887323, −9.068009341495681843581214566932, −8.215366034070726822555281767790, −8.025182240327055554162865718635, −7.60258845307742530676905928177, −7.31323916647783776161616920568, −7.02542777526991203048361344137, −6.51286214801338388695992446768, −5.52882351825582088230558306108, −5.26132808146977073078331134908, −5.07720104268189621158030876901, −5.01609126649749524861272647829, −3.89960281041964492089146350979, −3.64272914534569088635174482186, −3.05806191203817950983263147054, −2.42164441385462578075741634396, −1.59850011903782313259005445872, −1.54521801179125261400485037664, −0.43424280624175943825081437232,
0.43424280624175943825081437232, 1.54521801179125261400485037664, 1.59850011903782313259005445872, 2.42164441385462578075741634396, 3.05806191203817950983263147054, 3.64272914534569088635174482186, 3.89960281041964492089146350979, 5.01609126649749524861272647829, 5.07720104268189621158030876901, 5.26132808146977073078331134908, 5.52882351825582088230558306108, 6.51286214801338388695992446768, 7.02542777526991203048361344137, 7.31323916647783776161616920568, 7.60258845307742530676905928177, 8.025182240327055554162865718635, 8.215366034070726822555281767790, 9.068009341495681843581214566932, 9.348533940456258489932083887323, 9.491916162995338632485745474389
