| L(s) = 1 | − 4·5-s − 4·13-s − 20·17-s − 38·25-s − 52·29-s − 52·37-s − 116·41-s + 50·49-s − 148·53-s − 52·61-s + 16·65-s − 92·73-s + 80·85-s − 164·89-s + 4·97-s − 148·101-s + 92·109-s + 220·113-s + 194·121-s + 268·125-s + 127-s + 131-s + 137-s + 139-s + 208·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4/5·5-s − 0.307·13-s − 1.17·17-s − 1.51·25-s − 1.79·29-s − 1.40·37-s − 2.82·41-s + 1.02·49-s − 2.79·53-s − 0.852·61-s + 0.246·65-s − 1.26·73-s + 0.941·85-s − 1.84·89-s + 4/97·97-s − 1.46·101-s + 0.844·109-s + 1.94·113-s + 1.60·121-s + 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.43·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03353473845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03353473845\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−11.00171488376096661069126238030, −10.11721894220532623320476267332, −10.03339038021942265155617093094, −9.341573384592258607710989130980, −9.026804562832237329270032206298, −8.425752793492289960442165289762, −8.186931484919404454133865768759, −7.50058990697695686600016205770, −7.30454726657398001805928434464, −6.72783537549393540043726355964, −6.28407772241014722268335405618, −5.56255503334711989418259179337, −5.27728783387667732534606325156, −4.34201172346643658386682509998, −4.32682990891179909219678171999, −3.37548664029518989069451793032, −3.19551640293675521952493593074, −1.93233315050610312982951861936, −1.77008093139877958591002900531, −0.06602336337095114662774609823,
0.06602336337095114662774609823, 1.77008093139877958591002900531, 1.93233315050610312982951861936, 3.19551640293675521952493593074, 3.37548664029518989069451793032, 4.32682990891179909219678171999, 4.34201172346643658386682509998, 5.27728783387667732534606325156, 5.56255503334711989418259179337, 6.28407772241014722268335405618, 6.72783537549393540043726355964, 7.30454726657398001805928434464, 7.50058990697695686600016205770, 8.186931484919404454133865768759, 8.425752793492289960442165289762, 9.026804562832237329270032206298, 9.341573384592258607710989130980, 10.03339038021942265155617093094, 10.11721894220532623320476267332, 11.00171488376096661069126238030
