L(s) = 1 | − 9-s + 2·13-s + 17-s + 25-s − 49-s − 2·53-s + 81-s − 2·89-s + 2·101-s − 2·117-s + ⋯ |
L(s) = 1 | − 9-s + 2·13-s + 17-s + 25-s − 49-s − 2·53-s + 81-s − 2·89-s + 2·101-s − 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.066456845\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066456845\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.11607116313888964352218061541, −9.081802150278103874746541284819, −8.473192560861006315063926554132, −7.77307745955764230151618591072, −6.50780021151692892930741056642, −5.92395760898668046421472765951, −4.99491168444129834428163193342, −3.69379665731245283156746754868, −2.96339081761551840908085493334, −1.33674598123762693440695948513,
1.33674598123762693440695948513, 2.96339081761551840908085493334, 3.69379665731245283156746754868, 4.99491168444129834428163193342, 5.92395760898668046421472765951, 6.50780021151692892930741056642, 7.77307745955764230151618591072, 8.473192560861006315063926554132, 9.081802150278103874746541284819, 10.11607116313888964352218061541