| L(s) = 1 | + 3-s + 9-s + 8·11-s + 12·13-s − 8·23-s − 10·25-s + 27-s + 8·33-s + 4·37-s + 12·39-s + 24·47-s − 10·49-s + 8·59-s + 4·61-s − 8·69-s + 8·71-s − 20·73-s − 10·75-s + 81-s − 24·83-s − 12·97-s + 8·99-s + 24·107-s − 4·109-s + 4·111-s + 12·117-s + 26·121-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 2.41·11-s + 3.32·13-s − 1.66·23-s − 2·25-s + 0.192·27-s + 1.39·33-s + 0.657·37-s + 1.92·39-s + 3.50·47-s − 1.42·49-s + 1.04·59-s + 0.512·61-s − 0.963·69-s + 0.949·71-s − 2.34·73-s − 1.15·75-s + 1/9·81-s − 2.63·83-s − 1.21·97-s + 0.804·99-s + 2.32·107-s − 0.383·109-s + 0.379·111-s + 1.10·117-s + 2.36·121-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
| \(L(1)\) |
\(\approx\) |
\(3.321867930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.321867930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−8.658903060264415544055202217864, −8.396078230698006089481678801982, −7.70036027517747780731408099423, −7.31199108027851415208504538208, −6.52243750033071865852983380426, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −5.75568326693151721019291253980, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −3.94434944725762226728078180666, −3.36330701716196318281397292070, −2.30530051971351607466902904067, −1.50413098873776121963751941726, −1.16901968292393942798169217496,
1.16901968292393942798169217496, 1.50413098873776121963751941726, 2.30530051971351607466902904067, 3.36330701716196318281397292070, 3.94434944725762226728078180666, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 5.75568326693151721019291253980, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 6.52243750033071865852983380426, 7.31199108027851415208504538208, 7.70036027517747780731408099423, 8.396078230698006089481678801982, 8.658903060264415544055202217864
