L(s) = 1 | + 1.87·2-s + 3-s + 1.51·4-s − 1.33·5-s + 1.87·6-s − 0.902·8-s + 9-s − 2.49·10-s + 1.19·11-s + 1.51·12-s + 0.956·13-s − 1.33·15-s − 4.73·16-s + 3.26·17-s + 1.87·18-s − 6.16·19-s − 2.02·20-s + 2.23·22-s − 6.99·23-s − 0.902·24-s − 3.22·25-s + 1.79·26-s + 27-s − 2.28·29-s − 2.49·30-s − 1.84·31-s − 7.06·32-s + ⋯ |
L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.759·4-s − 0.595·5-s + 0.765·6-s − 0.318·8-s + 0.333·9-s − 0.789·10-s + 0.359·11-s + 0.438·12-s + 0.265·13-s − 0.343·15-s − 1.18·16-s + 0.791·17-s + 0.442·18-s − 1.41·19-s − 0.452·20-s + 0.476·22-s − 1.45·23-s − 0.184·24-s − 0.645·25-s + 0.351·26-s + 0.192·27-s − 0.424·29-s − 0.455·30-s − 0.331·31-s − 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 - 0.956T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 + 2.28T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 4.26T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.06T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 0.622T + 97T^{2} \) |
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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
−7.60421972685412005366042038361, −6.93777361951846633059664886595, −6.07293209687046148827799087246, −5.58893665359499562009787693826, −4.56277356366568181878386972857, −3.89803933464193831590958349693, −3.64378831461821399549200942364, −2.59256329527972157761309898118, −1.73808363524645536915559535152, 0,
1.73808363524645536915559535152, 2.59256329527972157761309898118, 3.64378831461821399549200942364, 3.89803933464193831590958349693, 4.56277356366568181878386972857, 5.58893665359499562009787693826, 6.07293209687046148827799087246, 6.93777361951846633059664886595, 7.60421972685412005366042038361