| L(s) = 1 | + 2·5-s + 2·7-s + 2·11-s − 6·17-s + 4·19-s − 2·23-s − 2·25-s + 12·29-s + 4·31-s + 4·35-s − 10·41-s + 8·43-s + 12·47-s + 3·49-s + 16·53-s + 4·55-s + 4·59-s + 16·67-s + 10·71-s − 12·73-s + 4·77-s + 8·79-s − 8·83-s − 12·85-s − 18·89-s + 8·95-s + 4·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.603·11-s − 1.45·17-s + 0.917·19-s − 0.417·23-s − 2/5·25-s + 2.22·29-s + 0.718·31-s + 0.676·35-s − 1.56·41-s + 1.21·43-s + 1.75·47-s + 3/7·49-s + 2.19·53-s + 0.539·55-s + 0.520·59-s + 1.95·67-s + 1.18·71-s − 1.40·73-s + 0.455·77-s + 0.900·79-s − 0.878·83-s − 1.30·85-s − 1.90·89-s + 0.820·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.809026527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.809026527\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.768137402244535847866796932088, −8.331059421386494329317535114470, −7.906569868728588213330358670629, −7.57060651306954935280200700049, −6.97760734989091340556664164026, −6.89979256518111088045281614264, −6.26412570113405014079556565276, −6.24878460444227487272859827261, −5.48413474621270504095013845841, −5.44847645574987078954552426945, −4.83313134233721954403699952949, −4.55178044882878232659814119914, −3.92996662271102959429877738833, −3.92624817712006397653975481362, −3.01352091743234589590375156062, −2.64653201428946196027156339575, −2.06166184904272634594278488980, −1.95318024269597081746576794008, −1.00424009298909905885542056993, −0.75788259247681868851266581062,
0.75788259247681868851266581062, 1.00424009298909905885542056993, 1.95318024269597081746576794008, 2.06166184904272634594278488980, 2.64653201428946196027156339575, 3.01352091743234589590375156062, 3.92624817712006397653975481362, 3.92996662271102959429877738833, 4.55178044882878232659814119914, 4.83313134233721954403699952949, 5.44847645574987078954552426945, 5.48413474621270504095013845841, 6.24878460444227487272859827261, 6.26412570113405014079556565276, 6.89979256518111088045281614264, 6.97760734989091340556664164026, 7.57060651306954935280200700049, 7.906569868728588213330358670629, 8.331059421386494329317535114470, 8.768137402244535847866796932088
