L(s) = 1 | + 2-s − 3.13·3-s + 4-s − 5-s − 3.13·6-s − 3.32·7-s + 8-s + 6.81·9-s − 10-s − 2.28·11-s − 3.13·12-s − 5.71·13-s − 3.32·14-s + 3.13·15-s + 16-s − 2.86·17-s + 6.81·18-s + 7.01·19-s − 20-s + 10.4·21-s − 2.28·22-s + 5.35·23-s − 3.13·24-s + 25-s − 5.71·26-s − 11.9·27-s − 3.32·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.80·3-s + 0.5·4-s − 0.447·5-s − 1.27·6-s − 1.25·7-s + 0.353·8-s + 2.27·9-s − 0.316·10-s − 0.688·11-s − 0.904·12-s − 1.58·13-s − 0.889·14-s + 0.808·15-s + 0.250·16-s − 0.695·17-s + 1.60·18-s + 1.61·19-s − 0.223·20-s + 2.27·21-s − 0.486·22-s + 1.11·23-s − 0.639·24-s + 0.200·25-s − 1.12·26-s − 2.30·27-s − 0.628·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 2.28T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 2.86T + 17T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 - 7.71T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 - 2.29T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 5.84T + 43T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 - 6.18T + 79T^{2} \) |
| 83 | \( 1 - 4.12T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 5.84T + 97T^{2} \) |
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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
−7.15908297367137073243705520233, −6.90299071134984144974710988927, −6.33567438307626403927939054630, −5.31542727177438501579891956061, −5.06191128645857582782449107001, −4.39435049455955815252785703473, −3.27924958693537781250407328354, −2.57536938800176124468426470164, −0.956832069712740317812697841386, 0,
0.956832069712740317812697841386, 2.57536938800176124468426470164, 3.27924958693537781250407328354, 4.39435049455955815252785703473, 5.06191128645857582782449107001, 5.31542727177438501579891956061, 6.33567438307626403927939054630, 6.90299071134984144974710988927, 7.15908297367137073243705520233