| L(s) = 1 | − 2·2-s + 3-s − 4-s − 2·5-s − 2·6-s + 8·8-s − 9-s + 4·10-s − 8·11-s − 12-s + 2·13-s − 2·15-s − 7·16-s + 17-s + 2·18-s − 2·19-s + 2·20-s + 16·22-s + 7·23-s + 8·24-s + 3·25-s − 4·26-s − 4·29-s + 4·30-s + 4·31-s − 14·32-s − 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.816·6-s + 2.82·8-s − 1/3·9-s + 1.26·10-s − 2.41·11-s − 0.288·12-s + 0.554·13-s − 0.516·15-s − 7/4·16-s + 0.242·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s + 3.41·22-s + 1.45·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.742·29-s + 0.730·30-s + 0.718·31-s − 2.47·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21669025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21669025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6937951414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6937951414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19 T + 168 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 154 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 120 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.499555800165976179090241420744, −8.304207052239372048829900108016, −7.85539142359360271925996600167, −7.62581382220344975068570348691, −7.37110865101491431901238503376, −7.11798838419446062429318334466, −6.40465201611480166397106496848, −5.88153830994060045801109274452, −5.49745071156399892259642633437, −5.00263599720633444537785896273, −4.77204802926732721452450569412, −4.54023692352780098850070326023, −3.78172405581200044164131864986, −3.69823844694640686355049469218, −2.98424535215597918787955064075, −2.63991311301045442091198520714, −2.17741958491259444849299433549, −1.40562025155029145531392977207, −0.62762076970209455963595117196, −0.52805726447003017788823012360,
0.52805726447003017788823012360, 0.62762076970209455963595117196, 1.40562025155029145531392977207, 2.17741958491259444849299433549, 2.63991311301045442091198520714, 2.98424535215597918787955064075, 3.69823844694640686355049469218, 3.78172405581200044164131864986, 4.54023692352780098850070326023, 4.77204802926732721452450569412, 5.00263599720633444537785896273, 5.49745071156399892259642633437, 5.88153830994060045801109274452, 6.40465201611480166397106496848, 7.11798838419446062429318334466, 7.37110865101491431901238503376, 7.62581382220344975068570348691, 7.85539142359360271925996600167, 8.304207052239372048829900108016, 8.499555800165976179090241420744
