| L(s) = 1 | + 12·13-s + 8·25-s − 32·31-s + 28·37-s + 16·43-s − 48·61-s − 32·67-s − 12·73-s + 20·97-s + 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 3.32·13-s + 8/5·25-s − 5.74·31-s + 4.60·37-s + 2.43·43-s − 6.14·61-s − 3.90·67-s − 1.40·73-s + 2.03·97-s + 1.57·103-s − 1.81·121-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 + 5.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.484896876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.484896876\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.422086396016260802238858819151, −7.71657750109353381973083640549, −7.69435875814692657276421279218, −7.65984530545156064177098316393, −7.57529067912904237890333773014, −7.15362759833428185811671736880, −6.75428581824896180174500678132, −6.33671650585931222341706671289, −6.15659129300509817029603398526, −5.92740473855369370401405931841, −5.87969600314991506112827772909, −5.65897159369838674556641163117, −5.35289135027668175713423740649, −4.65367612681756547262594712751, −4.44220781751072172516157745919, −4.36745726296670984490516728245, −3.99451668117529367565606951937, −3.70713420445550801069924603530, −3.07147214086244520619584582023, −3.04838393155607177083994596426, −3.00404297095822869316968961654, −1.93513938772314484534875481100, −1.73011782665181828286077374697, −1.35861733944311496550605413841, −0.70539636424688377879002508097,
0.70539636424688377879002508097, 1.35861733944311496550605413841, 1.73011782665181828286077374697, 1.93513938772314484534875481100, 3.00404297095822869316968961654, 3.04838393155607177083994596426, 3.07147214086244520619584582023, 3.70713420445550801069924603530, 3.99451668117529367565606951937, 4.36745726296670984490516728245, 4.44220781751072172516157745919, 4.65367612681756547262594712751, 5.35289135027668175713423740649, 5.65897159369838674556641163117, 5.87969600314991506112827772909, 5.92740473855369370401405931841, 6.15659129300509817029603398526, 6.33671650585931222341706671289, 6.75428581824896180174500678132, 7.15362759833428185811671736880, 7.57529067912904237890333773014, 7.65984530545156064177098316393, 7.69435875814692657276421279218, 7.71657750109353381973083640549, 8.422086396016260802238858819151
