L(s) = 1 | − 2.36·2-s − 1.16·3-s + 3.59·4-s − 3.78·5-s + 2.74·6-s − 1.24·7-s − 3.78·8-s − 1.65·9-s + 8.95·10-s + 5.90·11-s − 4.17·12-s − 1.18·13-s + 2.95·14-s + 4.39·15-s + 1.75·16-s + 17-s + 3.90·18-s − 1.27·19-s − 13.6·20-s + 1.45·21-s − 13.9·22-s − 7.24·23-s + 4.39·24-s + 9.32·25-s + 2.79·26-s + 5.40·27-s − 4.49·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 0.670·3-s + 1.79·4-s − 1.69·5-s + 1.12·6-s − 0.472·7-s − 1.33·8-s − 0.550·9-s + 2.83·10-s + 1.77·11-s − 1.20·12-s − 0.327·13-s + 0.790·14-s + 1.13·15-s + 0.438·16-s + 0.242·17-s + 0.921·18-s − 0.291·19-s − 3.04·20-s + 0.316·21-s − 2.97·22-s − 1.51·23-s + 0.896·24-s + 1.86·25-s + 0.548·26-s + 1.03·27-s − 0.849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 19 | \( 1 + 1.27T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 + 0.876T + 43T^{2} \) |
| 47 | \( 1 + 3.03T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 4.50T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.72026193253079980550471923928, −7.35467687237034528070544681717, −6.40291063744416953801127517175, −6.14408772215701111092214461057, −4.72418948328728175134330133148, −3.91683688837424391376299189315, −3.22116660574461081415219801851, −1.90638218151000654122743634472, −0.74975291948737585402022018519, 0,
0.74975291948737585402022018519, 1.90638218151000654122743634472, 3.22116660574461081415219801851, 3.91683688837424391376299189315, 4.72418948328728175134330133148, 6.14408772215701111092214461057, 6.40291063744416953801127517175, 7.35467687237034528070544681717, 7.72026193253079980550471923928