| L(s) = 1 | − 2·2-s − 4-s + 8·8-s − 2·11-s + 9·13-s − 7·16-s − 17-s − 2·19-s + 4·22-s − 18·26-s − 11·29-s + 10·31-s − 14·32-s + 2·34-s + 5·37-s + 4·38-s + 5·41-s + 2·43-s + 2·44-s + 6·47-s + 6·49-s − 9·52-s − 5·53-s + 22·58-s + 8·59-s + 61-s − 20·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s − 0.603·11-s + 2.49·13-s − 7/4·16-s − 0.242·17-s − 0.458·19-s + 0.852·22-s − 3.53·26-s − 2.04·29-s + 1.79·31-s − 2.47·32-s + 0.342·34-s + 0.821·37-s + 0.648·38-s + 0.780·41-s + 0.304·43-s + 0.301·44-s + 0.875·47-s + 6/7·49-s − 1.24·52-s − 0.686·53-s + 2.88·58-s + 1.04·59-s + 0.128·61-s − 2.54·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9893407207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9893407207\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 3 p T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 49 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 87 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 121 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 219 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 213 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.345552629124238242729169124274, −8.068879158137820640600228864461, −7.889608784936010687153318818565, −7.49489922456851570440298442812, −6.98220643525069411866824949071, −6.71450766438425203696794959947, −6.13285877543936065925075238617, −5.84573387921575067050587818701, −5.45103230119916507876157030511, −5.15415647227732069674956261999, −4.58895021170904813033880785010, −4.02223727241711605263513180881, −3.94668717888455358389982129262, −3.84105478798141704607848685073, −2.83876760396979048579855146444, −2.58856050799556273377415841197, −1.60373189184982908247302308115, −1.57278414419040865392828956051, −0.804413271596880927503481744996, −0.49476306090409125963806906411,
0.49476306090409125963806906411, 0.804413271596880927503481744996, 1.57278414419040865392828956051, 1.60373189184982908247302308115, 2.58856050799556273377415841197, 2.83876760396979048579855146444, 3.84105478798141704607848685073, 3.94668717888455358389982129262, 4.02223727241711605263513180881, 4.58895021170904813033880785010, 5.15415647227732069674956261999, 5.45103230119916507876157030511, 5.84573387921575067050587818701, 6.13285877543936065925075238617, 6.71450766438425203696794959947, 6.98220643525069411866824949071, 7.49489922456851570440298442812, 7.889608784936010687153318818565, 8.068879158137820640600228864461, 8.345552629124238242729169124274
