| L(s) = 1 | − 2·9-s + 8·29-s + 4·31-s − 8·41-s + 16·49-s − 8·61-s + 16·71-s − 8·79-s + 3·81-s + 8·89-s + 8·101-s + 16·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 1.48·29-s + 0.718·31-s − 1.24·41-s + 16/7·49-s − 1.02·61-s + 1.89·71-s − 0.900·79-s + 1/3·81-s + 0.847·89-s + 0.796·101-s + 1.53·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-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 + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5035099339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5035099339\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2898 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 2442 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 21138 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 152 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 9858 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 21558 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 176 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 14118 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.34568950735058480235066511636, −5.30764779361031412179393877869, −5.07457419250107867299805681852, −4.86166721912674569871549596122, −4.73926468989438961720249728887, −4.48471466711537505763047507469, −4.40055078120175865314570193057, −3.97707844917958713769105924308, −3.86194879505063908038540541221, −3.76587346667737063737597610900, −3.75352246127471995589823007061, −3.10742890774622215234316341793, −3.05798655101282022371662981803, −2.98948706098254082014379108805, −2.97062157519876391790613428701, −2.27410277279298980543621316776, −2.22741223180926970330921418773, −2.21255299633185449729806515224, −2.17832604468212772676523797417, −1.46741105569673774391405823946, −1.22457099824862640694992527357, −1.07514478311011423113272874551, −1.03500861272833061898140514337, −0.47449069399347014037836113511, −0.092723193434503146768041818815,
0.092723193434503146768041818815, 0.47449069399347014037836113511, 1.03500861272833061898140514337, 1.07514478311011423113272874551, 1.22457099824862640694992527357, 1.46741105569673774391405823946, 2.17832604468212772676523797417, 2.21255299633185449729806515224, 2.22741223180926970330921418773, 2.27410277279298980543621316776, 2.97062157519876391790613428701, 2.98948706098254082014379108805, 3.05798655101282022371662981803, 3.10742890774622215234316341793, 3.75352246127471995589823007061, 3.76587346667737063737597610900, 3.86194879505063908038540541221, 3.97707844917958713769105924308, 4.40055078120175865314570193057, 4.48471466711537505763047507469, 4.73926468989438961720249728887, 4.86166721912674569871549596122, 5.07457419250107867299805681852, 5.30764779361031412179393877869, 5.34568950735058480235066511636
