| 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 − 6·20-s + 4·22-s + 2·23-s + 3·25-s + 4·26-s − 6·28-s − 4·29-s + 6·32-s + 4·34-s + 4·35-s + 6·37-s − 8·40-s − 12·41-s + 8·43-s + 6·44-s + 4·46-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 − 1.34·20-s + 0.852·22-s + 0.417·23-s + 3/5·25-s + 0.784·26-s − 1.13·28-s − 0.742·29-s + 1.06·32-s + 0.685·34-s + 0.676·35-s + 0.986·37-s − 1.26·40-s − 1.87·41-s + 1.21·43-s + 0.904·44-s + 0.589·46-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
| \(L(1)\) |
\(\approx\) |
\(8.052664825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.052664825\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 146 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.88594916832782213722805779605, −7.72852876427675319310153161820, −7.21275183054847475942391177650, −7.09597438178479917992950539120, −6.66452054258638000614625511292, −6.34376795523346754118643219604, −5.88968874639855336546153818619, −5.79416072819502012093045576829, −5.21446452060390176500679630656, −4.94217470702336875362142336315, −4.45707599956264882745400231796, −4.10121632927673535516053437105, −3.66257760733444569967005570895, −3.64001199544440805396990302866, −3.09151955668516811345198854114, −2.79283853617632354207624939887, −2.11766708283321327541829782228, −1.82301941694625308049499587760, −0.882893546732091835435674859998, −0.68426280617406574793262842056,
0.68426280617406574793262842056, 0.882893546732091835435674859998, 1.82301941694625308049499587760, 2.11766708283321327541829782228, 2.79283853617632354207624939887, 3.09151955668516811345198854114, 3.64001199544440805396990302866, 3.66257760733444569967005570895, 4.10121632927673535516053437105, 4.45707599956264882745400231796, 4.94217470702336875362142336315, 5.21446452060390176500679630656, 5.79416072819502012093045576829, 5.88968874639855336546153818619, 6.34376795523346754118643219604, 6.66452054258638000614625511292, 7.09597438178479917992950539120, 7.21275183054847475942391177650, 7.72852876427675319310153161820, 7.88594916832782213722805779605
