| L(s) = 1 | − 2·7-s + 8·13-s − 4·19-s − 5·25-s + 4·31-s + 20·37-s + 10·43-s − 9·49-s + 14·61-s + 16·67-s + 14·73-s + 16·79-s − 16·91-s − 16·97-s + 52·103-s + 44·109-s − 17·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 2.21·13-s − 0.917·19-s − 25-s + 0.718·31-s + 3.28·37-s + 1.52·43-s − 9/7·49-s + 1.79·61-s + 1.95·67-s + 1.63·73-s + 1.80·79-s − 1.67·91-s − 1.62·97-s + 5.12·103-s + 4.21·109-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.056257453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.056257453\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 48 T^{4} + p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 17 T^{2} + 240 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 53 T^{2} + 1272 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 2310 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 7326 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 89 T^{2} + 5400 T^{4} + 89 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 28 T^{2} + 5286 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 6966 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7 T + 60 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 + 92 T^{2} + 3750 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 188 T^{2} + 17862 T^{4} + 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 116 T^{2} + 18678 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.29676364964237424845507769060, −5.91481280872669157378273828487, −5.90250817810239269958784095901, −5.77308214461059582812985593409, −5.73449531872430911726024473638, −5.11589735546545637990026916461, −5.07160538779094083545479267602, −4.79611241091220360371901940495, −4.56426651941744798108376417970, −4.20762262891916088886630148617, −4.17683027324391254597584021923, −3.97823690290518255640872029998, −3.80885592016021356528968144328, −3.33642056655243582922104858778, −3.26369476884850995450017055975, −3.17966453059115101880052687183, −2.90123731454787858272763175881, −2.33466469563871227316883924625, −2.05855310072684263868301924778, −2.05019797840654913141762739331, −2.04243174127748300706522723755, −1.14202793708184089471615663433, −0.956996296094860209534034619459, −0.77047052557336047411496168979, −0.47879729343034893960514321785,
0.47879729343034893960514321785, 0.77047052557336047411496168979, 0.956996296094860209534034619459, 1.14202793708184089471615663433, 2.04243174127748300706522723755, 2.05019797840654913141762739331, 2.05855310072684263868301924778, 2.33466469563871227316883924625, 2.90123731454787858272763175881, 3.17966453059115101880052687183, 3.26369476884850995450017055975, 3.33642056655243582922104858778, 3.80885592016021356528968144328, 3.97823690290518255640872029998, 4.17683027324391254597584021923, 4.20762262891916088886630148617, 4.56426651941744798108376417970, 4.79611241091220360371901940495, 5.07160538779094083545479267602, 5.11589735546545637990026916461, 5.73449531872430911726024473638, 5.77308214461059582812985593409, 5.90250817810239269958784095901, 5.91481280872669157378273828487, 6.29676364964237424845507769060
