L(s) = 1 | − 9·3-s − 71·7-s + 81·9-s − 191·13-s − 601·19-s + 639·21-s − 729·27-s + 1.55e3·31-s + 2.06e3·37-s + 1.71e3·39-s − 3.19e3·43-s + 2.64e3·49-s + 5.40e3·57-s + 7.19e3·61-s − 5.75e3·63-s + 8.80e3·67-s + 8.54e3·73-s + 7.68e3·79-s + 6.56e3·81-s + 1.35e4·91-s − 1.40e4·93-s − 9.07e3·97-s − 1.64e4·103-s − 1.87e4·109-s − 1.85e4·111-s − 1.54e4·117-s + ⋯ |
L(s) = 1 | − 3-s − 1.44·7-s + 9-s − 1.13·13-s − 1.66·19-s + 1.44·21-s − 27-s + 1.62·31-s + 1.50·37-s + 1.13·39-s − 1.72·43-s + 1.09·49-s + 1.66·57-s + 1.93·61-s − 1.44·63-s + 1.96·67-s + 1.60·73-s + 1.23·79-s + 81-s + 1.63·91-s − 1.62·93-s − 0.964·97-s − 1.54·103-s − 1.57·109-s − 1.50·111-s − 1.13·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6726011457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6726011457\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 71 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 191 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 + 601 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 - 1559 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2062 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 3191 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 7199 T + p^{4} T^{2} \) |
| 67 | \( 1 - 8809 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 8542 T + p^{4} T^{2} \) |
| 79 | \( 1 - 7682 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 + 9071 T + p^{4} T^{2} \) |
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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
−11.09983734630539392727539559543, −10.03624016075824611514470287854, −9.653110739828543612089462821927, −8.164087184168340754654666060898, −6.75670464424963822729722870302, −6.39779787011724144417460469437, −5.10568194631890160253944543146, −3.98968107275729611086561496948, −2.45615454895435937753756402893, −0.50786256077308963817178164574,
0.50786256077308963817178164574, 2.45615454895435937753756402893, 3.98968107275729611086561496948, 5.10568194631890160253944543146, 6.39779787011724144417460469437, 6.75670464424963822729722870302, 8.164087184168340754654666060898, 9.653110739828543612089462821927, 10.03624016075824611514470287854, 11.09983734630539392727539559543