| L(s) = 1 | + 53·9-s − 90·11-s − 182·19-s − 288·29-s − 52·31-s − 918·41-s + 10·49-s − 144·59-s − 236·61-s − 216·71-s − 1.79e3·79-s + 2.08e3·81-s − 702·89-s − 4.77e3·99-s + 1.60e3·109-s + 3.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.45e3·169-s − 9.64e3·171-s + ⋯ |
| L(s) = 1 | + 1.96·9-s − 2.46·11-s − 2.19·19-s − 1.84·29-s − 0.301·31-s − 3.49·41-s + 0.0291·49-s − 0.317·59-s − 0.495·61-s − 0.361·71-s − 2.55·79-s + 2.85·81-s − 0.836·89-s − 4.84·99-s + 1.40·109-s + 2.56·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.11·169-s − 4.31·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4501183010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4501183010\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 45 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2458 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3863 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 26 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 459 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 52586 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 11378 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 13610 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 538525 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 688633 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 898 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 284245 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 351 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1676350 T^{2} + p^{6} T^{4} \) |
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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
−11.42719545365814315039984594559, −10.56274284429775148549756514350, −10.22952993191318345558407880096, −9.922413046763944970091042655067, −9.389966774357262107591964086596, −8.555030750358521281633070061985, −8.427460416979262564376503711780, −7.67919442404474539100835531862, −7.48651968093925570464671610454, −6.82256444712194132480159139523, −6.57221334238163972215349957233, −5.55708454945625139759102586587, −5.39723636120386282890234038006, −4.54329257044329439848044059741, −4.35926437920469053597212026933, −3.55242601428833158527110280667, −2.85166632103282400950275483469, −1.85758253460488688080849070393, −1.79458506374998210883146754356, −0.21101499771311045873645691305,
0.21101499771311045873645691305, 1.79458506374998210883146754356, 1.85758253460488688080849070393, 2.85166632103282400950275483469, 3.55242601428833158527110280667, 4.35926437920469053597212026933, 4.54329257044329439848044059741, 5.39723636120386282890234038006, 5.55708454945625139759102586587, 6.57221334238163972215349957233, 6.82256444712194132480159139523, 7.48651968093925570464671610454, 7.67919442404474539100835531862, 8.427460416979262564376503711780, 8.555030750358521281633070061985, 9.389966774357262107591964086596, 9.922413046763944970091042655067, 10.22952993191318345558407880096, 10.56274284429775148549756514350, 11.42719545365814315039984594559
