| L(s) = 1 | + 2-s − 7-s − 8-s + 3·9-s + 2·11-s − 14-s − 16-s − 7·17-s + 3·18-s + 2·22-s + 3·23-s + 12·29-s + 7·31-s − 7·34-s − 4·37-s − 14·41-s + 16·43-s + 3·46-s − 7·47-s − 6·49-s + 4·53-s + 56-s + 12·58-s + 14·59-s + 14·61-s + 7·62-s − 3·63-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s + 9-s + 0.603·11-s − 0.267·14-s − 1/4·16-s − 1.69·17-s + 0.707·18-s + 0.426·22-s + 0.625·23-s + 2.22·29-s + 1.25·31-s − 1.20·34-s − 0.657·37-s − 2.18·41-s + 2.43·43-s + 0.442·46-s − 1.02·47-s − 6/7·49-s + 0.549·53-s + 0.133·56-s + 1.57·58-s + 1.82·59-s + 1.79·61-s + 0.889·62-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.217063219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217063219\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−11.86829101099404790262712064987, −11.30186564908633296258363199411, −10.91873840622636396596505910858, −10.26103845395232419857560481787, −9.828613902065009203468454290904, −9.656785970563092861908815034313, −8.675316991609847082404949933469, −8.646716636315804039807457408765, −8.116192448671307693468838603486, −7.07327476270978309141219495726, −6.80991542849221230349033387897, −6.60121912851246192474476526756, −5.96132472153781885865701835896, −4.96516076979752092931154537741, −4.88489845734171203037522675899, −4.10702267231352623105337212255, −3.76230548808931036220762617250, −2.85248647170399003682824530588, −2.20496407227810848497295250088, −1.00315219492347088514687749877,
1.00315219492347088514687749877, 2.20496407227810848497295250088, 2.85248647170399003682824530588, 3.76230548808931036220762617250, 4.10702267231352623105337212255, 4.88489845734171203037522675899, 4.96516076979752092931154537741, 5.96132472153781885865701835896, 6.60121912851246192474476526756, 6.80991542849221230349033387897, 7.07327476270978309141219495726, 8.116192448671307693468838603486, 8.646716636315804039807457408765, 8.675316991609847082404949933469, 9.656785970563092861908815034313, 9.828613902065009203468454290904, 10.26103845395232419857560481787, 10.91873840622636396596505910858, 11.30186564908633296258363199411, 11.86829101099404790262712064987
