L(s) = 1 | + 2-s − 2.80·3-s + 4-s − 2.44·5-s − 2.80·6-s + 0.0489·7-s + 8-s + 4.85·9-s − 2.44·10-s − 5.80·11-s − 2.80·12-s − 4.44·13-s + 0.0489·14-s + 6.85·15-s + 16-s − 2.55·17-s + 4.85·18-s − 4.26·19-s − 2.44·20-s − 0.137·21-s − 5.80·22-s − 4.69·23-s − 2.80·24-s + 0.978·25-s − 4.44·26-s − 5.18·27-s + 0.0489·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.61·3-s + 0.5·4-s − 1.09·5-s − 1.14·6-s + 0.0184·7-s + 0.353·8-s + 1.61·9-s − 0.773·10-s − 1.74·11-s − 0.808·12-s − 1.23·13-s + 0.0130·14-s + 1.76·15-s + 0.250·16-s − 0.619·17-s + 1.14·18-s − 0.979·19-s − 0.546·20-s − 0.0299·21-s − 1.23·22-s − 0.978·23-s − 0.571·24-s + 0.195·25-s − 0.871·26-s − 0.998·27-s + 0.00924·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3011 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 0.0489T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 + 4.44T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 4.26T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 - 6.89T + 31T^{2} \) |
| 37 | \( 1 + 4.89T + 37T^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 - 0.423T + 53T^{2} \) |
| 59 | \( 1 - 0.417T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 0.0827T + 71T^{2} \) |
| 73 | \( 1 - 2.91T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 - 5.42T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + 5.87T + 97T^{2} \) |
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
−7.17955607849783625532295179694, −6.55980525995023381034083768659, −5.83866748283632855031560313658, −4.92460973584157887601387916627, −4.83031749984607127365507291763, −3.96440069559322117728101305879, −2.88113922792060225793490564596, −1.90387730588182140982630630979, 0, 0,
1.90387730588182140982630630979, 2.88113922792060225793490564596, 3.96440069559322117728101305879, 4.83031749984607127365507291763, 4.92460973584157887601387916627, 5.83866748283632855031560313658, 6.55980525995023381034083768659, 7.17955607849783625532295179694