L(s) = 1 | + 4·5-s + 4·13-s + 12·17-s + 2·25-s + 20·37-s + 4·41-s + 49-s − 24·53-s + 12·61-s + 16·65-s − 4·73-s + 48·85-s − 12·89-s − 36·97-s + 28·101-s + 4·109-s − 32·113-s − 18·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 1.10·13-s + 2.91·17-s + 2/5·25-s + 3.28·37-s + 0.624·41-s + 1/7·49-s − 3.29·53-s + 1.53·61-s + 1.98·65-s − 0.468·73-s + 5.20·85-s − 1.27·89-s − 3.65·97-s + 2.78·101-s + 0.383·109-s − 3.01·113-s − 1.63·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.432973797\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.432973797\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−8.348093232759484287064194714284, −8.016710564482448925343947507213, −7.79064855464535275520223516342, −7.16759940438630821447644837255, −6.32147524527224537125089512133, −6.18921559649042192845851641510, −5.73569194086517270564919024713, −5.50301216544551056360312768488, −4.88053771103235633099685381109, −4.10502926185056205242132098716, −3.60698435135707065201863764202, −2.88871942698849130206456913367, −2.45683901738953215093540180120, −1.38744136941764418419874219848, −1.23073197938907604520302484273,
1.23073197938907604520302484273, 1.38744136941764418419874219848, 2.45683901738953215093540180120, 2.88871942698849130206456913367, 3.60698435135707065201863764202, 4.10502926185056205242132098716, 4.88053771103235633099685381109, 5.50301216544551056360312768488, 5.73569194086517270564919024713, 6.18921559649042192845851641510, 6.32147524527224537125089512133, 7.16759940438630821447644837255, 7.79064855464535275520223516342, 8.016710564482448925343947507213, 8.348093232759484287064194714284