L(s) = 1 | + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)7-s + (0.846 + 1.75i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)28-s − 1.56i·31-s + (−0.777 − 0.974i)37-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (−1.90 − 0.433i)52-s + (−1.22 + 0.974i)61-s + (0.900 + 0.433i)64-s − 0.445·67-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)4-s + (0.900 − 0.433i)7-s + (0.846 + 1.75i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (−0.900 − 0.433i)25-s + (−0.222 + 0.974i)28-s − 1.56i·31-s + (−0.777 − 0.974i)37-s + (−0.400 − 1.75i)43-s + (0.623 − 0.781i)49-s + (−1.90 − 0.433i)52-s + (−1.22 + 0.974i)61-s + (0.900 + 0.433i)64-s − 0.445·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8199486964\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8199486964\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.900 + 0.433i)T \) |
good | 2 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 5 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 - 0.867iT - T^{2} \) |
| 23 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 29 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + 1.56iT - T^{2} \) |
| 37 | \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 61 | \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + 0.445T + T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 + 0.867iT - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.64364771649385788898897980470, −10.60004763095687692657772975245, −9.449664008636115201962183406063, −8.661773684538616129423888503632, −7.87604119782695562181828581310, −6.98701937484244941042677298668, −5.67292758131999990757562775390, −4.29872827720491438378431383270, −3.86018323134596211897653580691, −1.93272760017737609322555567365,
1.41866505169058123650939135130, 3.21070083437703102155310474852, 4.74069340594935601604084111127, 5.40500684954348814411753277659, 6.34730658325799405366575058053, 7.85414118753343608850092718838, 8.535151236509782505486619123309, 9.421416255214919497228291777971, 10.50935424216346858335191192172, 10.99291433140716464507877548440