| L(s) = 1 | + 4-s + 9-s + 11-s + 16-s − 17-s − 23-s + 25-s − 31-s + 36-s − 37-s − 41-s − 43-s + 44-s + 49-s + 2·61-s + 64-s − 68-s − 71-s − 73-s − 79-s + 81-s + 2·83-s − 89-s − 92-s + 99-s + 100-s − 113-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s + 11-s + 16-s − 17-s − 23-s + 25-s − 31-s + 36-s − 37-s − 41-s − 43-s + 44-s + 49-s + 2·61-s + 64-s − 68-s − 71-s − 73-s − 79-s + 81-s + 2·83-s − 89-s − 92-s + 99-s + 100-s − 113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2563 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.704728343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704728343\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 233 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−9.020009334974855580080227092243, −8.347176406388379027602919459600, −7.23446647040270288310065959559, −6.89934430819586723881572927115, −6.22587729353245931673220085239, −5.20827325966413228781924762719, −4.17336978762595771348959176719, −3.44817324685758271131182164983, −2.18914478852139758103090914772, −1.43634349865073593314751131417,
1.43634349865073593314751131417, 2.18914478852139758103090914772, 3.44817324685758271131182164983, 4.17336978762595771348959176719, 5.20827325966413228781924762719, 6.22587729353245931673220085239, 6.89934430819586723881572927115, 7.23446647040270288310065959559, 8.347176406388379027602919459600, 9.020009334974855580080227092243
