L(s) = 1 | − 2-s + 1.15·3-s + 4-s − 5-s − 1.15·6-s − 7-s − 8-s − 1.65·9-s + 10-s + 1.15·12-s + 3.58·13-s + 14-s − 1.15·15-s + 16-s − 4.29·17-s + 1.65·18-s + 4.89·19-s − 20-s − 1.15·21-s + 2.85·23-s − 1.15·24-s + 25-s − 3.58·26-s − 5.39·27-s − 28-s + 0.754·29-s + 1.15·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.668·3-s + 0.5·4-s − 0.447·5-s − 0.472·6-s − 0.377·7-s − 0.353·8-s − 0.552·9-s + 0.316·10-s + 0.334·12-s + 0.994·13-s + 0.267·14-s − 0.299·15-s + 0.250·16-s − 1.04·17-s + 0.391·18-s + 1.12·19-s − 0.223·20-s − 0.252·21-s + 0.595·23-s − 0.236·24-s + 0.200·25-s − 0.703·26-s − 1.03·27-s − 0.188·28-s + 0.140·29-s + 0.211·30-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
\(L(1)\) |
\(\approx\) |
\(1.275808394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.275808394\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.15T + 3T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 2.85T + 23T^{2} \) |
| 29 | \( 1 - 0.754T + 29T^{2} \) |
| 31 | \( 1 + 9.35T + 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 - 0.0689T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 - 4.03T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 - 9.06T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−7.84241623361564645705644268865, −7.27323021573178991280398578432, −6.63650510367661599105275629766, −5.80889124403628298704986929429, −5.12390135781779566746589285402, −3.89244420145374662174819399356, −3.42300869797423318968291142603, −2.64977687368417988506175998044, −1.75016932311114465513670479677, −0.58551640801742939130316041430,
0.58551640801742939130316041430, 1.75016932311114465513670479677, 2.64977687368417988506175998044, 3.42300869797423318968291142603, 3.89244420145374662174819399356, 5.12390135781779566746589285402, 5.80889124403628298704986929429, 6.63650510367661599105275629766, 7.27323021573178991280398578432, 7.84241623361564645705644268865