| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 3·8-s + 3·9-s − 10-s − 2·12-s + 4·13-s − 2·15-s − 16-s + 3·18-s + 20-s − 6·24-s + 25-s + 4·26-s + 4·27-s − 2·30-s + 5·32-s − 3·36-s + 20·37-s + 8·39-s + 3·40-s + 20·41-s − 8·43-s − 3·45-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 1.06·8-s + 9-s − 0.316·10-s − 0.577·12-s + 1.10·13-s − 0.516·15-s − 1/4·16-s + 0.707·18-s + 0.223·20-s − 1.22·24-s + 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.365·30-s + 0.883·32-s − 1/2·36-s + 3.28·37-s + 1.28·39-s + 0.474·40-s + 3.12·41-s − 1.21·43-s − 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481361971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481361971\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.907104671360377962786893632766, −9.080341213166513123570518780286, −8.843956595114680827452383410006, −8.438902886313495638742074033009, −7.68003844385765985083943135134, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −5.86497653842965752491328535183, −5.85130665991850657402735978605, −4.52160444884874669934496061646, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −3.01549784140686353642765162463, −2.63252540175087701435373844808, −1.18815390030287753587945796282,
1.18815390030287753587945796282, 2.63252540175087701435373844808, 3.01549784140686353642765162463, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 4.52160444884874669934496061646, 5.85130665991850657402735978605, 5.86497653842965752491328535183, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 7.68003844385765985083943135134, 8.438902886313495638742074033009, 8.843956595114680827452383410006, 9.080341213166513123570518780286, 9.907104671360377962786893632766
