| L(s) = 1 | − 8·13-s − 8·25-s + 4·37-s + 28·49-s − 20·61-s + 32·73-s + 32·97-s + 40·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.21·13-s − 8/5·25-s + 0.657·37-s + 4·49-s − 2.56·61-s + 3.74·73-s + 3.24·97-s + 3.83·109-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 + 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.032255162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.032255162\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^3$ | \( 1 - 40 T^{2} + 759 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 160 T^{2} + 17679 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−5.90512282240284304788229481496, −5.63927154070331116753211035818, −5.23515749897040817216075076434, −5.21406900061405236171211109271, −4.96753054076817479872080553494, −4.89273390192100788110266274406, −4.63927313223451585412329548999, −4.54165886810893450161982865652, −4.23092302942697922284976631043, −3.98413890076781687611828487419, −3.77821495235021711603103446633, −3.56514181126161561209769695487, −3.53247572780904680312551956047, −3.32509242252547628569842056586, −2.70807755751182065342865537785, −2.53877794328209561406103702826, −2.52852978264933874074830824163, −2.45219871092086943389723068281, −2.10956459053755008440120579878, −1.83714056424128204172559856913, −1.52243726918188566356411789082, −1.28159682950827740947064428673, −0.76418202422969752001674906828, −0.53854489454888031335303204672, −0.31683963366681137534308895202,
0.31683963366681137534308895202, 0.53854489454888031335303204672, 0.76418202422969752001674906828, 1.28159682950827740947064428673, 1.52243726918188566356411789082, 1.83714056424128204172559856913, 2.10956459053755008440120579878, 2.45219871092086943389723068281, 2.52852978264933874074830824163, 2.53877794328209561406103702826, 2.70807755751182065342865537785, 3.32509242252547628569842056586, 3.53247572780904680312551956047, 3.56514181126161561209769695487, 3.77821495235021711603103446633, 3.98413890076781687611828487419, 4.23092302942697922284976631043, 4.54165886810893450161982865652, 4.63927313223451585412329548999, 4.89273390192100788110266274406, 4.96753054076817479872080553494, 5.21406900061405236171211109271, 5.23515749897040817216075076434, 5.63927154070331116753211035818, 5.90512282240284304788229481496
