| L(s) = 1 | − 96·25-s − 196·49-s + 384·73-s − 576·97-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 476·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 3.83·25-s − 4·49-s + 5.26·73-s − 5.93·97-s − 4·121-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 − 2.81·169-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 + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7315148350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7315148350\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2}( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 1680 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 5040 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2}( 1 + 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 12480 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.18218578877772472100235833310, −6.08263385305934633375010398408, −5.82287184795361258802538952150, −5.43256611014626679866277313907, −5.31985803890213966103242703215, −5.23448687442913221682203019954, −5.16711941158549230838030334824, −4.64865654372599771548220061868, −4.56352456407870083070616293950, −4.17156508503479391566961883679, −3.97562738636550127968618913005, −3.88345147126213382359122231364, −3.73368506844486739178244947320, −3.37290479873308110697482871740, −3.22188906562215573384401901285, −2.76277005879972610911122811248, −2.72425374233045724991112439856, −2.21143227908311375317370987045, −2.18194333067412668526138470071, −1.76567182193755470798264271207, −1.60670615511753637354622513128, −1.31365068187300220348025650684, −0.968329767209690045799321167933, −0.34607726208268549180599819722, −0.16319160761093751898636623169,
0.16319160761093751898636623169, 0.34607726208268549180599819722, 0.968329767209690045799321167933, 1.31365068187300220348025650684, 1.60670615511753637354622513128, 1.76567182193755470798264271207, 2.18194333067412668526138470071, 2.21143227908311375317370987045, 2.72425374233045724991112439856, 2.76277005879972610911122811248, 3.22188906562215573384401901285, 3.37290479873308110697482871740, 3.73368506844486739178244947320, 3.88345147126213382359122231364, 3.97562738636550127968618913005, 4.17156508503479391566961883679, 4.56352456407870083070616293950, 4.64865654372599771548220061868, 5.16711941158549230838030334824, 5.23448687442913221682203019954, 5.31985803890213966103242703215, 5.43256611014626679866277313907, 5.82287184795361258802538952150, 6.08263385305934633375010398408, 6.18218578877772472100235833310
