L(s) = 1 | + 2-s + 1.14·3-s + 4-s + 5-s + 1.14·6-s + 7-s + 8-s − 1.68·9-s + 10-s + 1.14·12-s + 14-s + 1.14·15-s + 16-s − 2.68·17-s − 1.68·18-s + 5.53·19-s + 20-s + 1.14·21-s − 1.83·23-s + 1.14·24-s + 25-s − 5.37·27-s + 28-s + 3.83·29-s + 1.14·30-s + 9.66·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.661·3-s + 0.5·4-s + 0.447·5-s + 0.468·6-s + 0.377·7-s + 0.353·8-s − 0.561·9-s + 0.316·10-s + 0.330·12-s + 0.267·14-s + 0.295·15-s + 0.250·16-s − 0.651·17-s − 0.397·18-s + 1.27·19-s + 0.223·20-s + 0.250·21-s − 0.382·23-s + 0.234·24-s + 0.200·25-s − 1.03·27-s + 0.188·28-s + 0.711·29-s + 0.209·30-s + 1.73·31-s + 0.176·32-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
\(L(1)\) |
\(\approx\) |
\(4.896851939\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.896851939\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.14T + 3T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 5.53T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 - 4.97T + 43T^{2} \) |
| 47 | \( 1 + 0.393T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 0.685T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 0.585T + 71T^{2} \) |
| 73 | \( 1 + 8.97T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 9.37T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.74158337071030412704895476804, −7.10370111792033627629832926597, −6.27435434691687912970198636475, −5.68757453185233107641659273540, −4.97655566258029294224382771120, −4.27832068138821671020105535832, −3.39147442546807728218434604590, −2.69746653862613125457301363187, −2.10200146146536315585326498236, −0.959195220023369098798078773193,
0.959195220023369098798078773193, 2.10200146146536315585326498236, 2.69746653862613125457301363187, 3.39147442546807728218434604590, 4.27832068138821671020105535832, 4.97655566258029294224382771120, 5.68757453185233107641659273540, 6.27435434691687912970198636475, 7.10370111792033627629832926597, 7.74158337071030412704895476804