L(s) = 1 | + 2.15·2-s − 2.86·3-s + 2.66·4-s − 1.37·5-s − 6.18·6-s + 7-s + 1.43·8-s + 5.19·9-s − 2.96·10-s + 11-s − 7.62·12-s − 13-s + 2.15·14-s + 3.92·15-s − 2.23·16-s − 1.89·17-s + 11.2·18-s − 4.96·19-s − 3.65·20-s − 2.86·21-s + 2.15·22-s − 3.02·23-s − 4.10·24-s − 3.12·25-s − 2.15·26-s − 6.27·27-s + 2.66·28-s + ⋯ |
L(s) = 1 | + 1.52·2-s − 1.65·3-s + 1.33·4-s − 0.613·5-s − 2.52·6-s + 0.377·7-s + 0.506·8-s + 1.73·9-s − 0.936·10-s + 0.301·11-s − 2.20·12-s − 0.277·13-s + 0.577·14-s + 1.01·15-s − 0.558·16-s − 0.460·17-s + 2.64·18-s − 1.13·19-s − 0.816·20-s − 0.624·21-s + 0.460·22-s − 0.631·23-s − 0.836·24-s − 0.624·25-s − 0.423·26-s − 1.20·27-s + 0.503·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 + 6.60T + 41T^{2} \) |
| 43 | \( 1 + 8.33T + 43T^{2} \) |
| 47 | \( 1 - 3.15T + 47T^{2} \) |
| 53 | \( 1 - 0.902T + 53T^{2} \) |
| 59 | \( 1 - 8.95T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 2.35T + 73T^{2} \) |
| 79 | \( 1 - 3.27T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−9.970997798137872070468514448707, −8.622017779074711887006928085365, −7.36570430965960144298092064799, −6.61594369087924349207047974997, −5.88961368576673480497661480905, −5.16462399514261481102032568935, −4.37018698379217062519172618241, −3.76143242183743953309578378130, −2.02408000818825048138268196027, 0,
2.02408000818825048138268196027, 3.76143242183743953309578378130, 4.37018698379217062519172618241, 5.16462399514261481102032568935, 5.88961368576673480497661480905, 6.61594369087924349207047974997, 7.36570430965960144298092064799, 8.622017779074711887006928085365, 9.970997798137872070468514448707