L(s) = 1 | − 3-s + 4·7-s + 9-s + 12·13-s − 4·21-s − 10·25-s − 27-s + 20·31-s + 4·37-s − 12·39-s + 16·43-s − 2·49-s + 4·61-s + 4·63-s + 8·67-s − 20·73-s + 10·75-s − 12·79-s + 81-s + 48·91-s − 20·93-s − 12·97-s − 20·103-s − 4·109-s − 4·111-s + 12·117-s − 6·121-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 3.32·13-s − 0.872·21-s − 2·25-s − 0.192·27-s + 3.59·31-s + 0.657·37-s − 1.92·39-s + 2.43·43-s − 2/7·49-s + 0.512·61-s + 0.503·63-s + 0.977·67-s − 2.34·73-s + 1.15·75-s − 1.35·79-s + 1/9·81-s + 5.03·91-s − 2.07·93-s − 1.21·97-s − 1.97·103-s − 0.383·109-s − 0.379·111-s + 1.10·117-s − 0.545·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.478326877\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.478326877\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.396078230698006089481678801982, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −7.31199108027851415208504538208, −6.40139715881748782453956361829, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −3.12662782741322079682173901720, −2.30530051971351607466902904067, −1.27109672072212401682622254955, −1.16901968292393942798169217496,
1.16901968292393942798169217496, 1.27109672072212401682622254955, 2.30530051971351607466902904067, 3.12662782741322079682173901720, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 6.40139715881748782453956361829, 7.31199108027851415208504538208, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 8.396078230698006089481678801982