| L(s) = 1 | − 2-s + 4-s − 0.416·5-s + 4.65·7-s − 8-s + 0.416·10-s + 0.283·13-s − 4.65·14-s + 16-s − 0.640·17-s − 5.88·19-s − 0.416·20-s − 5.88·23-s − 4.82·25-s − 0.283·26-s + 4.65·28-s − 5.16·29-s + 1.22·31-s − 32-s + 0.640·34-s − 1.94·35-s − 6.82·37-s + 5.88·38-s + 0.416·40-s − 12.4·41-s − 3.07·43-s + 5.88·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.186·5-s + 1.76·7-s − 0.353·8-s + 0.131·10-s + 0.0785·13-s − 1.24·14-s + 0.250·16-s − 0.155·17-s − 1.35·19-s − 0.0931·20-s − 1.22·23-s − 0.965·25-s − 0.0555·26-s + 0.880·28-s − 0.959·29-s + 0.220·31-s − 0.176·32-s + 0.109·34-s − 0.328·35-s − 1.12·37-s + 0.954·38-s + 0.0658·40-s − 1.93·41-s − 0.469·43-s + 0.867·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.416T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.283T + 13T^{2} \) |
| 17 | \( 1 + 0.640T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 + 5.16T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 3.07T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 + 3.20T + 59T^{2} \) |
| 61 | \( 1 + 1.16T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 + 9.27T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 - 8.99T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.372266957406235639648891212811, −7.84363447788108101083878577672, −7.06408484829827677000675224261, −6.15093957558764154903876995077, −5.27656393659833719421544388174, −4.45504665392988890756299492150, −3.60326075697916086481872890048, −2.06384254901616198896140584986, −1.68683543102571850675880246447, 0,
1.68683543102571850675880246447, 2.06384254901616198896140584986, 3.60326075697916086481872890048, 4.45504665392988890756299492150, 5.27656393659833719421544388174, 6.15093957558764154903876995077, 7.06408484829827677000675224261, 7.84363447788108101083878577672, 8.372266957406235639648891212811
