| L(s) = 1 | + 48·3-s + 1.59e3·9-s − 2.76e3·11-s − 5.30e3·17-s − 2.01e4·19-s + 1.29e4·25-s + 8.07e4·27-s − 1.32e5·33-s + 2.09e5·41-s − 1.16e5·43-s + 2.83e5·49-s − 2.54e5·51-s − 9.65e5·57-s − 8.94e5·59-s − 9.87e5·67-s − 2.19e5·73-s + 6.21e5·75-s + 2.67e6·81-s − 5.44e4·83-s + 3.72e6·89-s + 2.91e6·97-s − 4.41e6·99-s − 2.92e6·107-s − 1.10e5·113-s − 1.40e6·121-s + 1.00e7·123-s + ⋯ |
| L(s) = 1 | + 16/9·3-s + 2.18·9-s − 2.07·11-s − 1.07·17-s − 2.93·19-s + 0.829·25-s + 4.10·27-s − 3.69·33-s + 3.04·41-s − 1.46·43-s + 2.40·49-s − 1.91·51-s − 5.21·57-s − 4.35·59-s − 3.28·67-s − 0.564·73-s + 1.47·75-s + 5.02·81-s − 0.0951·83-s + 5.28·89-s + 3.19·97-s − 4.55·99-s − 2.38·107-s − 0.0765·113-s − 0.795·121-s + 5.41·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4451070271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4451070271\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - 8 p T + 22 p T^{2} - 8 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12956 T^{2} + 20439126 p^{2} T^{4} - 12956 p^{12} T^{6} + p^{24} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 283076 T^{2} + 45562329030 T^{4} - 283076 p^{12} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 1384 T + 3578082 T^{2} + 1384 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8625884 T^{2} + 53582343042726 T^{4} - 8625884 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 156 p T + 74822 p^{2} T^{2} + 156 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10056 T + 95372546 T^{2} + 10056 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 335576132 T^{2} + 111239326161126 p^{2} T^{4} - 335576132 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 704504924 T^{2} + 163770654668790 p^{2} T^{4} - 704504924 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 608707844 T^{2} + 1230450287881000710 T^{4} - 608707844 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 5901139612 T^{2} + 17163805463118788454 T^{4} - 5901139612 p^{12} T^{6} + p^{24} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 104924 T + 12138830502 T^{2} - 104924 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 58408 T + 232079910 p T^{2} + 58408 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 36633896324 T^{2} + \)\(55\!\cdots\!26\)\( T^{4} - 36633896324 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 86160561436 T^{2} + \)\(28\!\cdots\!90\)\( T^{4} - 86160561436 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 447304 T + 128510653986 T^{2} + 447304 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 131358477404 T^{2} + \)\(91\!\cdots\!90\)\( T^{4} - 131358477404 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 493544 T + 216456943938 T^{2} + 493544 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 399213568964 T^{2} + \)\(72\!\cdots\!30\)\( T^{4} - 399213568964 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 109764 T + 128374575206 T^{2} + 109764 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 691585720964 T^{2} + \)\(23\!\cdots\!30\)\( T^{4} - 691585720964 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 27208 T + 404800477698 T^{2} + 27208 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 1861500 T + 1842501984422 T^{2} - 1861500 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 1459172 T + 1653043299654 T^{2} - 1459172 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−7.79680181886021966390151586050, −7.38687941154751441608152317490, −7.35803536112275390991379017749, −6.92585254266002714279446623709, −6.56669526715099703348915804554, −6.50112816979013301337246998974, −6.03117875142580621676197384945, −5.87848242140938450208885948890, −5.59330388505684489026190098384, −4.97715889920023008265343124981, −4.61571718551264554816129019840, −4.42453819480886603720108426043, −4.40924151151587955730634638419, −4.38424293751647697918565588177, −3.41164469763660828301806103448, −3.26171066802670981283651099175, −3.13029759278365549684827570358, −2.66868973021685771431835726803, −2.28342529015102557418217955744, −2.19208899178517823929466567952, −2.09058915763488323716753325477, −1.42679877506258174201580861230, −1.01132917359906854116457908149, −0.60605968163710023699374893698, −0.06819147467122113175911184348,
0.06819147467122113175911184348, 0.60605968163710023699374893698, 1.01132917359906854116457908149, 1.42679877506258174201580861230, 2.09058915763488323716753325477, 2.19208899178517823929466567952, 2.28342529015102557418217955744, 2.66868973021685771431835726803, 3.13029759278365549684827570358, 3.26171066802670981283651099175, 3.41164469763660828301806103448, 4.38424293751647697918565588177, 4.40924151151587955730634638419, 4.42453819480886603720108426043, 4.61571718551264554816129019840, 4.97715889920023008265343124981, 5.59330388505684489026190098384, 5.87848242140938450208885948890, 6.03117875142580621676197384945, 6.50112816979013301337246998974, 6.56669526715099703348915804554, 6.92585254266002714279446623709, 7.35803536112275390991379017749, 7.38687941154751441608152317490, 7.79680181886021966390151586050
