| L(s) = 1 | + 48·9-s + 704·25-s + 1.79e3·49-s − 2.14e3·73-s + 900·81-s − 7.74e3·97-s − 1.93e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.19e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 3.37e4·225-s + ⋯ |
| L(s) = 1 | + 16/9·9-s + 5.63·25-s + 5.22·49-s − 3.43·73-s + 1.23·81-s − 8.10·97-s − 14.5·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 5.42·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 10.0·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.28400736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.28400736\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 8 p T^{2} + 46 p^{2} T^{4} - 8 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| good | 5 | \( ( 1 - 176 T^{2} + 38862 T^{4} - 176 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 7 | \( ( 1 - 64 p T^{2} + 215646 T^{4} - 64 p^{7} T^{6} + p^{12} T^{8} )^{4} \) |
| 11 | \( ( 1 + 40 p^{2} T^{2} + 849210 p T^{4} + 40 p^{8} T^{6} + p^{12} T^{8} )^{4} \) |
| 13 | \( ( 1 + 2980 T^{2} + 4014966 T^{4} + 2980 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 17 | \( ( 1 - 17188 T^{2} + 120704262 T^{4} - 17188 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 19 | \( ( 1 - 968 p T^{2} + 170208990 T^{4} - 968 p^{7} T^{6} + p^{12} T^{8} )^{4} \) |
| 23 | \( ( 1 + 31196 T^{2} + 464722854 T^{4} + 31196 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 29 | \( ( 1 - 53936 T^{2} + 1889351598 T^{4} - 53936 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 31 | \( ( 1 - 98176 T^{2} + 4116020094 T^{4} - 98176 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 37 | \( ( 1 + 124996 T^{2} + 9025572822 T^{4} + 124996 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 41 | \( ( 1 - 82084 T^{2} + 4965540774 T^{4} - 82084 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 43 | \( ( 1 - 216920 T^{2} + 24333819006 T^{4} - 216920 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 47 | \( ( 1 + 93500 T^{2} - 1715710650 T^{4} + 93500 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 53 | \( ( 1 - 418352 T^{2} + 87270813582 T^{4} - 418352 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 59 | \( ( 1 + 799912 T^{2} + 244209482046 T^{4} + 799912 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 61 | \( ( 1 + 357220 T^{2} + 77943572022 T^{4} + 357220 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 67 | \( ( 1 - 1127768 T^{2} + 498508041246 T^{4} - 1127768 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 71 | \( ( 1 + 776732 T^{2} + 403415373030 T^{4} + 776732 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 73 | \( ( 1 + 268 T + 73158 T^{2} + 268 p^{3} T^{3} + p^{6} T^{4} )^{8} \) |
| 79 | \( ( 1 - 943744 T^{2} + 544083847614 T^{4} - 943744 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 83 | \( ( 1 + 1890664 T^{2} + 1510361878110 T^{4} + 1890664 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 89 | \( ( 1 - 175300 T^{2} - 599506777050 T^{4} - 175300 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 97 | \( ( 1 + 968 T + 1792302 T^{2} + 968 p^{3} T^{3} + p^{6} T^{4} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.32812892808824409678525741454, −2.27774316857459171877405923779, −2.12575231712422555022841663854, −2.02003676395044706494872784929, −1.90416111509478735654101918874, −1.75169915692875587724823231098, −1.57691169510906748220474190613, −1.56413745422656271919421389573, −1.51153248951795696886085706190, −1.37052888985915160635550098946, −1.37048912549430991727549974720, −1.34562031547591704251366817335, −1.24531359451687012420526035248, −1.19701741596782240758939810376, −1.19090681722754319208214306478, −1.02893784143410224575982375712, −0.881423747350510720158108826551, −0.812896034102074443314750133584, −0.68876693458414558413892640944, −0.56523739811198617275368463496, −0.46091939412000654744980856011, −0.35276721974031741346572582080, −0.27964117418282957629836963721, −0.22448505202775333244703556915, −0.07175676264635778032992320772,
0.07175676264635778032992320772, 0.22448505202775333244703556915, 0.27964117418282957629836963721, 0.35276721974031741346572582080, 0.46091939412000654744980856011, 0.56523739811198617275368463496, 0.68876693458414558413892640944, 0.812896034102074443314750133584, 0.881423747350510720158108826551, 1.02893784143410224575982375712, 1.19090681722754319208214306478, 1.19701741596782240758939810376, 1.24531359451687012420526035248, 1.34562031547591704251366817335, 1.37048912549430991727549974720, 1.37052888985915160635550098946, 1.51153248951795696886085706190, 1.56413745422656271919421389573, 1.57691169510906748220474190613, 1.75169915692875587724823231098, 1.90416111509478735654101918874, 2.02003676395044706494872784929, 2.12575231712422555022841663854, 2.27774316857459171877405923779, 2.32812892808824409678525741454
Plot not available for L-functions of degree greater than 10.
