| L(s) = 1 | + 2-s − 2.53·3-s + 4-s − 5-s − 2.53·6-s − 7-s + 8-s + 3.44·9-s − 10-s − 2.53·12-s − 4.11·13-s − 14-s + 2.53·15-s + 16-s − 6.84·17-s + 3.44·18-s − 6.62·19-s − 20-s + 2.53·21-s − 3.13·23-s − 2.53·24-s + 25-s − 4.11·26-s − 1.13·27-s − 28-s + 5.49·29-s + 2.53·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.447·5-s − 1.03·6-s − 0.377·7-s + 0.353·8-s + 1.14·9-s − 0.316·10-s − 0.733·12-s − 1.14·13-s − 0.267·14-s + 0.655·15-s + 0.250·16-s − 1.66·17-s + 0.812·18-s − 1.51·19-s − 0.223·20-s + 0.554·21-s − 0.654·23-s − 0.518·24-s + 0.200·25-s − 0.806·26-s − 0.219·27-s − 0.188·28-s + 1.02·29-s + 0.463·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3956396380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3956396380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 - 2.90T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 0.0608T + 71T^{2} \) |
| 73 | \( 1 + 3.91T + 73T^{2} \) |
| 79 | \( 1 + 4.23T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 + 0.691T + 97T^{2} \) |
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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
−7.46300497608780960993241303012, −6.70052887534161112314846506227, −6.47772336946103016708727101341, −5.72441356458521956394307144254, −4.84566756246616418538584114006, −4.55213567574086037029855236028, −3.79155090573326289255349787236, −2.64921991915006968329801242433, −1.84342289878337857735896068073, −0.28039423432335704463269953369,
0.28039423432335704463269953369, 1.84342289878337857735896068073, 2.64921991915006968329801242433, 3.79155090573326289255349787236, 4.55213567574086037029855236028, 4.84566756246616418538584114006, 5.72441356458521956394307144254, 6.47772336946103016708727101341, 6.70052887534161112314846506227, 7.46300497608780960993241303012
