L(s) = 1 | + 2-s + 4-s + 1.72·5-s − 2.60·7-s + 8-s + 1.72·10-s − 5.87·11-s + 3.97·13-s − 2.60·14-s + 16-s + 3.69·17-s − 0.972·19-s + 1.72·20-s − 5.87·22-s − 1.45·23-s − 2.02·25-s + 3.97·26-s − 2.60·28-s + 0.793·29-s − 5.90·31-s + 32-s + 3.69·34-s − 4.49·35-s − 1.91·37-s − 0.972·38-s + 1.72·40-s + 2.77·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.771·5-s − 0.984·7-s + 0.353·8-s + 0.545·10-s − 1.77·11-s + 1.10·13-s − 0.696·14-s + 0.250·16-s + 0.894·17-s − 0.223·19-s + 0.385·20-s − 1.25·22-s − 0.302·23-s − 0.404·25-s + 0.779·26-s − 0.492·28-s + 0.147·29-s − 1.06·31-s + 0.176·32-s + 0.632·34-s − 0.759·35-s − 0.314·37-s − 0.157·38-s + 0.272·40-s + 0.434·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 + 2.60T + 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 13 | \( 1 - 3.97T + 13T^{2} \) |
| 17 | \( 1 - 3.69T + 17T^{2} \) |
| 19 | \( 1 + 0.972T + 19T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 - 0.793T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 + 1.91T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 - 0.964T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 + 8.91T + 61T^{2} \) |
| 67 | \( 1 - 1.13T + 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 + 6.18T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 2.08T + 89T^{2} \) |
| 97 | \( 1 - 4.60T + 97T^{2} \) |
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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
−7.51391630321469010795255334002, −6.46569175820769368957518210720, −6.04659513592741483519610030788, −5.47884244358887686446545055227, −4.83785527167539463032988493053, −3.71710371978438042404892417336, −3.18047784287463671012766992736, −2.41680377049469375870152238854, −1.50336970869346827549236182273, 0,
1.50336970869346827549236182273, 2.41680377049469375870152238854, 3.18047784287463671012766992736, 3.71710371978438042404892417336, 4.83785527167539463032988493053, 5.47884244358887686446545055227, 6.04659513592741483519610030788, 6.46569175820769368957518210720, 7.51391630321469010795255334002