| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s − 4·7-s − 3·8-s + 3·9-s − 6·11-s − 4·12-s − 2·13-s − 4·14-s + 16-s − 4·17-s + 3·18-s + 2·19-s − 8·21-s − 6·22-s + 8·23-s − 6·24-s − 2·26-s + 4·27-s + 8·28-s − 10·29-s − 16·31-s + 2·32-s − 12·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s − 1.80·11-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.458·19-s − 1.74·21-s − 1.27·22-s + 1.66·23-s − 1.22·24-s − 0.392·26-s + 0.769·27-s + 1.51·28-s − 1.85·29-s − 2.87·31-s + 0.353·32-s − 2.08·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 111 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 161 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.181261381722060537884896846990, −9.173304439313308628419290032566, −8.601167175697993215128839140713, −8.384583687218046965842645639274, −7.41646744640811361960018894209, −7.33813706161780314131634873777, −7.23174901412969987237917317217, −6.55001549925351331738774475117, −5.67736581263467345663484127626, −5.60716530995683553534752010767, −5.02413268808905679111821291713, −4.80169256947718492348971663024, −4.05826716928951410698780780993, −3.57756546733495541605894688168, −3.37605293531069060471817792327, −2.87898590573163790275443095093, −2.33127289349717856914364225476, −1.68439519556190185103794196011, 0, 0,
1.68439519556190185103794196011, 2.33127289349717856914364225476, 2.87898590573163790275443095093, 3.37605293531069060471817792327, 3.57756546733495541605894688168, 4.05826716928951410698780780993, 4.80169256947718492348971663024, 5.02413268808905679111821291713, 5.60716530995683553534752010767, 5.67736581263467345663484127626, 6.55001549925351331738774475117, 7.23174901412969987237917317217, 7.33813706161780314131634873777, 7.41646744640811361960018894209, 8.384583687218046965842645639274, 8.601167175697993215128839140713, 9.173304439313308628419290032566, 9.181261381722060537884896846990
