| L(s) = 1 | − 2-s + 7-s + 8-s + 9-s + 2·11-s − 14-s − 16-s − 18-s − 2·22-s − 23-s + 25-s − 29-s − 37-s − 43-s + 46-s + 49-s − 50-s − 53-s + 56-s + 58-s + 63-s + 64-s + 2·67-s − 71-s + 72-s + 74-s + 2·77-s + ⋯ |
| L(s) = 1 | − 2-s + 7-s + 8-s + 9-s + 2·11-s − 14-s − 16-s − 18-s − 2·22-s − 23-s + 25-s − 29-s − 37-s − 43-s + 46-s + 49-s − 50-s − 53-s + 56-s + 58-s + 63-s + 64-s + 2·67-s − 71-s + 72-s + 74-s + 2·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8707474261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8707474261\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.274775205462396086911924458599, −8.661890690620726942400588362753, −7.946283262814151194489326617650, −7.12883486363224070013170047113, −6.54070800839605145928991104821, −5.21652005024942019894680999878, −4.34250585188019629694596369731, −3.77319252512681084685192193105, −1.81547097199917722662223656197, −1.28315484536059843848231174569,
1.28315484536059843848231174569, 1.81547097199917722662223656197, 3.77319252512681084685192193105, 4.34250585188019629694596369731, 5.21652005024942019894680999878, 6.54070800839605145928991104821, 7.12883486363224070013170047113, 7.946283262814151194489326617650, 8.661890690620726942400588362753, 9.274775205462396086911924458599
