| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 2·7-s − 4·8-s + 4·10-s + 2·11-s + 2·13-s − 4·14-s + 5·16-s + 2·17-s − 2·19-s − 6·20-s − 4·22-s + 2·23-s + 3·25-s − 4·26-s + 6·28-s − 6·29-s − 2·31-s − 6·32-s − 4·34-s − 4·35-s − 10·37-s + 4·38-s + 8·40-s + 8·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 1.26·10-s + 0.603·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s − 0.458·19-s − 1.34·20-s − 0.852·22-s + 0.417·23-s + 3/5·25-s − 0.784·26-s + 1.13·28-s − 1.11·29-s − 0.359·31-s − 1.06·32-s − 0.685·34-s − 0.676·35-s − 1.64·37-s + 0.648·38-s + 1.26·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550992940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550992940\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.209138036577532036905858507158, −7.892961283148553959244126926667, −7.34509548500927298650638117878, −7.29139060002777885667740653338, −6.92392220281245299589842728777, −6.69453948694384171829446367106, −5.92645620444322973267413253685, −5.88919343236635472487912923110, −5.31661741133760244222797054077, −5.14292359918261585565323250484, −4.36029373569550882188310309169, −4.09521224339112970532338909234, −3.65688024572109456142497806499, −3.53503284104572076127560360694, −2.65756782535004570359436589912, −2.50435420563622874003141740351, −1.80107209787269710138502592812, −1.49939834996934654036041795730, −0.820588338996863150286434266125, −0.52829117305309699357891062227,
0.52829117305309699357891062227, 0.820588338996863150286434266125, 1.49939834996934654036041795730, 1.80107209787269710138502592812, 2.50435420563622874003141740351, 2.65756782535004570359436589912, 3.53503284104572076127560360694, 3.65688024572109456142497806499, 4.09521224339112970532338909234, 4.36029373569550882188310309169, 5.14292359918261585565323250484, 5.31661741133760244222797054077, 5.88919343236635472487912923110, 5.92645620444322973267413253685, 6.69453948694384171829446367106, 6.92392220281245299589842728777, 7.29139060002777885667740653338, 7.34509548500927298650638117878, 7.892961283148553959244126926667, 8.209138036577532036905858507158
