| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·11-s + 6·12-s − 6·13-s + 4·14-s + 5·16-s − 2·17-s − 6·18-s − 4·19-s − 4·21-s − 8·22-s − 8·23-s − 8·24-s + 12·26-s + 4·27-s − 6·28-s − 8·29-s − 4·31-s − 6·32-s + 8·33-s + 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.20·11-s + 1.73·12-s − 1.66·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.917·19-s − 0.872·21-s − 1.70·22-s − 1.66·23-s − 1.63·24-s + 2.35·26-s + 0.769·27-s − 1.13·28-s − 1.48·29-s − 0.718·31-s − 1.06·32-s + 1.39·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 22 T + 262 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 174 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.752677768457262890497191805121, −8.337485126059074815061240004163, −8.120301109314120931509754123838, −7.66829346390298021077226010320, −7.23257998762657334452949963470, −6.96686921318522211159659429652, −6.63098031149746496571831001938, −6.32745162273406458023427021759, −5.68829218510361101543219965577, −5.34981077663953021974146322552, −4.47738089346060040263144933860, −4.26921732194890432187040349064, −3.54270894406654313486798345220, −3.45854097395211246013378726282, −2.63839662389258357743615785747, −2.33188837955964504683728384855, −1.73080267571195772374368158473, −1.52521759800200905206610807774, 0, 0,
1.52521759800200905206610807774, 1.73080267571195772374368158473, 2.33188837955964504683728384855, 2.63839662389258357743615785747, 3.45854097395211246013378726282, 3.54270894406654313486798345220, 4.26921732194890432187040349064, 4.47738089346060040263144933860, 5.34981077663953021974146322552, 5.68829218510361101543219965577, 6.32745162273406458023427021759, 6.63098031149746496571831001938, 6.96686921318522211159659429652, 7.23257998762657334452949963470, 7.66829346390298021077226010320, 8.120301109314120931509754123838, 8.337485126059074815061240004163, 8.752677768457262890497191805121
