| L(s) = 1 | + 16·19-s + 88·25-s − 224·43-s − 172·49-s − 112·67-s − 80·73-s − 32·97-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 460·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 0.842·19-s + 3.51·25-s − 5.20·43-s − 3.51·49-s − 1.67·67-s − 1.09·73-s − 0.329·97-s + 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.498710990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498710990\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 484 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 470 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2480 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 362 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 572 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3406 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2642 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 7178 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 7190 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 7946 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12800 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
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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
−6.19066372536253067241377633689, −6.08938886927974881332495745903, −5.91894020242106430473074646766, −5.38421577444179141327609495358, −5.25849218435432718568825888695, −5.21391203758571193131283406822, −4.85798600739912092335760058433, −4.69667679687453910184354155086, −4.63086595128369122463020202901, −4.60532605393144891630334899004, −4.10034167787061942483635509115, −3.49606872515066236519825376901, −3.47550252188685458212256777944, −3.41044219146073083077993268584, −3.34607045172856210343207018198, −2.85790340954569303426290853647, −2.73852871166277990876506453901, −2.37352115358092387477042796130, −2.20654581668163893283750854843, −1.59756060849291164150617677584, −1.37387186671812639405674641663, −1.27775596972124152608831929878, −1.23671756793364779417381779923, −0.37618521451779055772731825940, −0.20548878055316926007119508575,
0.20548878055316926007119508575, 0.37618521451779055772731825940, 1.23671756793364779417381779923, 1.27775596972124152608831929878, 1.37387186671812639405674641663, 1.59756060849291164150617677584, 2.20654581668163893283750854843, 2.37352115358092387477042796130, 2.73852871166277990876506453901, 2.85790340954569303426290853647, 3.34607045172856210343207018198, 3.41044219146073083077993268584, 3.47550252188685458212256777944, 3.49606872515066236519825376901, 4.10034167787061942483635509115, 4.60532605393144891630334899004, 4.63086595128369122463020202901, 4.69667679687453910184354155086, 4.85798600739912092335760058433, 5.21391203758571193131283406822, 5.25849218435432718568825888695, 5.38421577444179141327609495358, 5.91894020242106430473074646766, 6.08938886927974881332495745903, 6.19066372536253067241377633689
