| L(s) = 1 | + 2-s + 4-s + 2.45·5-s − 2.79·7-s + 8-s + 2.45·10-s + 1.67·11-s − 6.34·13-s − 2.79·14-s + 16-s − 4.96·17-s + 2.45·20-s + 1.67·22-s + 2.49·23-s + 1.04·25-s − 6.34·26-s − 2.79·28-s + 5.93·29-s − 7.28·31-s + 32-s − 4.96·34-s − 6.87·35-s − 0.550·37-s + 2.45·40-s − 2.60·41-s + 2.87·43-s + 1.67·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.09·5-s − 1.05·7-s + 0.353·8-s + 0.777·10-s + 0.506·11-s − 1.75·13-s − 0.746·14-s + 0.250·16-s − 1.20·17-s + 0.549·20-s + 0.357·22-s + 0.521·23-s + 0.209·25-s − 1.24·26-s − 0.527·28-s + 1.10·29-s − 1.30·31-s + 0.176·32-s − 0.852·34-s − 1.16·35-s − 0.0905·37-s + 0.388·40-s − 0.407·41-s + 0.437·43-s + 0.253·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 + 0.550T + 37T^{2} \) |
| 41 | \( 1 + 2.60T + 41T^{2} \) |
| 43 | \( 1 - 2.87T + 43T^{2} \) |
| 47 | \( 1 + 0.745T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 6.99T + 71T^{2} \) |
| 73 | \( 1 + 6.18T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 6.90T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.20504875751812594920515118640, −6.90081796469646558184736069121, −6.21148289170705503689053160992, −5.55609199825331104834776819147, −4.81978486454185974911812521647, −4.11800749622177607619107723984, −3.02334779670049144815492758934, −2.49302301228318736122441474513, −1.63264129442964048229248642330, 0,
1.63264129442964048229248642330, 2.49302301228318736122441474513, 3.02334779670049144815492758934, 4.11800749622177607619107723984, 4.81978486454185974911812521647, 5.55609199825331104834776819147, 6.21148289170705503689053160992, 6.90081796469646558184736069121, 7.20504875751812594920515118640
