| L(s) = 1 | + 1.22e3·23-s − 496·25-s − 1.80e3·47-s + 1.25e3·49-s + 432·71-s + 344·73-s + 1.24e3·97-s + 3.80e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.31e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 11.0·23-s − 3.96·25-s − 5.58·47-s + 3.65·49-s + 0.722·71-s + 0.551·73-s + 1.29·97-s + 2.86·121-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 + 5.98·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(22.88877745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.88877745\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 248 T^{2} + 1806 p^{2} T^{4} + 248 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 - 626 T^{2} + 327363 T^{4} - 626 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 1904 T^{2} + 3668622 T^{4} - 1904 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 3287 T^{2} + p^{6} T^{4} )^{4} \) |
| 17 | \( ( 1 - 5072 T^{2} + 1982238 T^{4} - 5072 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 19 | \( ( 1 + 18094 T^{2} + 171635955 T^{4} + 18094 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 306 T + 44422 T^{2} - 306 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 41866 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 - 116204 T^{2} + 5150826150 T^{4} - 116204 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 92182 T^{2} + 7044936315 T^{4} - 92182 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 30740 T^{2} + 8341944198 T^{4} - 30740 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 43 | \( ( 1 + 65524 T^{2} + 6829649142 T^{4} + 65524 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 450 T + 95542 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 + 154436 T^{2} + 4597192086 T^{4} + 154436 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 512528 T^{2} + 126780934542 T^{4} - 512528 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 763582 T^{2} + 246326003187 T^{4} - 763582 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + 743782 T^{2} + 274440697755 T^{4} + 743782 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 - 108 T + 599182 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - 86 T + 65499 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 - 1357322 T^{2} + 905419894299 T^{4} - 1357322 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 1580396 T^{2} + 1168518522966 T^{4} - 1580396 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - 2136560 T^{2} + 2018716991358 T^{4} - 2136560 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 310 T + 926871 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.41121994027112339912629064660, −3.32503182007761241035154099608, −3.30927200281234981102744210465, −3.16687904596614077791396530747, −2.98701743487187095477337310004, −2.97130191888838003336740596965, −2.90203036056044933062166099152, −2.68399754911891455890774556358, −2.58206503618285120451842799106, −2.40277729285197839174835094367, −2.20523520721050160490247364008, −2.07521186517975615602216602730, −1.91891384588047210690602733452, −1.89533617226338844741548794462, −1.58854002979042939330576217709, −1.48430273003261156442458109383, −1.34490550852125030756225633462, −1.09275151291812079428923444340, −1.05584059251955115483383213723, −1.05514191658952967580723594423, −0.69550262795321320430317543057, −0.59156762371269290862235023669, −0.55243817258086469057510520913, −0.38627805363634017043011227296, −0.16277547213795581016096174112,
0.16277547213795581016096174112, 0.38627805363634017043011227296, 0.55243817258086469057510520913, 0.59156762371269290862235023669, 0.69550262795321320430317543057, 1.05514191658952967580723594423, 1.05584059251955115483383213723, 1.09275151291812079428923444340, 1.34490550852125030756225633462, 1.48430273003261156442458109383, 1.58854002979042939330576217709, 1.89533617226338844741548794462, 1.91891384588047210690602733452, 2.07521186517975615602216602730, 2.20523520721050160490247364008, 2.40277729285197839174835094367, 2.58206503618285120451842799106, 2.68399754911891455890774556358, 2.90203036056044933062166099152, 2.97130191888838003336740596965, 2.98701743487187095477337310004, 3.16687904596614077791396530747, 3.30927200281234981102744210465, 3.32503182007761241035154099608, 3.41121994027112339912629064660
Plot not available for L-functions of degree greater than 10.
