L(s) = 1 | − 0.618i·2-s + (−0.221 + 0.221i)3-s + 0.618·4-s + (0.136 + 0.136i)6-s + (0.642 + 0.642i)7-s − i·8-s + 0.902i·9-s + (−0.136 + 0.136i)12-s + (0.396 − 0.396i)14-s + (−0.809 + 0.587i)17-s + 0.557·18-s − 0.284·21-s + (0.221 + 0.221i)24-s − i·25-s + (−0.420 − 0.420i)27-s + (0.396 + 0.396i)28-s + ⋯ |
L(s) = 1 | − 0.618i·2-s + (−0.221 + 0.221i)3-s + 0.618·4-s + (0.136 + 0.136i)6-s + (0.642 + 0.642i)7-s − i·8-s + 0.902i·9-s + (−0.136 + 0.136i)12-s + (0.396 − 0.396i)14-s + (−0.809 + 0.587i)17-s + 0.557·18-s − 0.284·21-s + (0.221 + 0.221i)24-s − i·25-s + (−0.420 − 0.420i)27-s + (0.396 + 0.396i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.125631769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125631769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (0.809 - 0.587i)T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 + 0.618iT - T^{2} \) |
| 3 | \( 1 + (0.221 - 0.221i)T - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (-0.642 - 0.642i)T + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.221 + 0.221i)T - iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.90iT - T^{2} \) |
| 59 | \( 1 + 0.618iT - T^{2} \) |
| 61 | \( 1 + (-1.26 - 1.26i)T + iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.39 - 1.39i)T - iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (1.39 + 1.39i)T + iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 + (1.26 - 1.26i)T - iT^{2} \) |
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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
−10.49338648165342073824202504039, −9.966639437166362958372919549371, −8.704147811499830345188998814028, −8.008839545527945066520295060311, −6.93869849391645339547969744342, −6.00162603757293823615768167264, −5.01079727752070267972009760957, −4.00011311388167499768079791635, −2.58582575141961639504201372891, −1.81603116458268595366090047535,
1.50021595385170185476727381398, 2.97073178173103081062108242035, 4.31041590630269292513213728810, 5.38267649122223876536397204972, 6.32585072265563079375187314641, 7.05680841141372253064926999143, 7.66824906152998017168633918940, 8.665242920766217113457990276462, 9.574257244015017322923054505536, 10.73444008164134370887114169788