| L(s) = 1 | − 2.20·2-s − 3-s + 2.88·4-s − 2.30·5-s + 2.20·6-s − 2.22·7-s − 1.94·8-s + 9-s + 5.10·10-s − 3.84·11-s − 2.88·12-s − 1.58·13-s + 4.90·14-s + 2.30·15-s − 1.45·16-s + 17-s − 2.20·18-s − 0.738·19-s − 6.65·20-s + 2.22·21-s + 8.49·22-s − 4.94·23-s + 1.94·24-s + 0.329·25-s + 3.49·26-s − 27-s − 6.39·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.44·4-s − 1.03·5-s + 0.902·6-s − 0.839·7-s − 0.688·8-s + 0.333·9-s + 1.61·10-s − 1.15·11-s − 0.831·12-s − 0.438·13-s + 1.31·14-s + 0.596·15-s − 0.364·16-s + 0.242·17-s − 0.520·18-s − 0.169·19-s − 1.48·20-s + 0.484·21-s + 1.81·22-s − 1.03·23-s + 0.397·24-s + 0.0658·25-s + 0.685·26-s − 0.192·27-s − 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 2.20T + 2T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 19 | \( 1 + 0.738T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 0.840T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 - 0.805T + 41T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 - 0.966T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 1.20T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + 4.59T + 73T^{2} \) |
| 79 | \( 1 - 7.33T + 79T^{2} \) |
| 83 | \( 1 + 0.714T + 83T^{2} \) |
| 89 | \( 1 + 3.74T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.73643897748469530051306760425, −6.90415000010739219735710534542, −6.54992366028342398330677746963, −5.49768016429460747023460397523, −4.72501442624806506525610667262, −3.81346763298742980781265835915, −2.89245856934402762633471324907, −1.99470866357523652301375159538, −0.66941560165384970895744357691, 0,
0.66941560165384970895744357691, 1.99470866357523652301375159538, 2.89245856934402762633471324907, 3.81346763298742980781265835915, 4.72501442624806506525610667262, 5.49768016429460747023460397523, 6.54992366028342398330677746963, 6.90415000010739219735710534542, 7.73643897748469530051306760425
