L(s) = 1 | + 2-s + 1.36·3-s + 4-s − 0.474·5-s + 1.36·6-s + 0.790·7-s + 8-s − 1.13·9-s − 0.474·10-s − 5.89·11-s + 1.36·12-s + 2.95·13-s + 0.790·14-s − 0.647·15-s + 16-s − 6.75·17-s − 1.13·18-s − 1.17·19-s − 0.474·20-s + 1.08·21-s − 5.89·22-s − 7.14·23-s + 1.36·24-s − 4.77·25-s + 2.95·26-s − 5.64·27-s + 0.790·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.789·3-s + 0.5·4-s − 0.212·5-s + 0.557·6-s + 0.298·7-s + 0.353·8-s − 0.377·9-s − 0.149·10-s − 1.77·11-s + 0.394·12-s + 0.820·13-s + 0.211·14-s − 0.167·15-s + 0.250·16-s − 1.63·17-s − 0.266·18-s − 0.269·19-s − 0.106·20-s + 0.235·21-s − 1.25·22-s − 1.48·23-s + 0.278·24-s − 0.955·25-s + 0.580·26-s − 1.08·27-s + 0.149·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 + 0.474T + 5T^{2} \) |
| 7 | \( 1 - 0.790T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 7.14T + 23T^{2} \) |
| 29 | \( 1 - 3.67T + 29T^{2} \) |
| 31 | \( 1 - 7.59T + 31T^{2} \) |
| 37 | \( 1 + 2.84T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 - 3.36T + 43T^{2} \) |
| 47 | \( 1 + 5.90T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 2.91T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6.16T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 9.71T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−8.046869739368987952116016597995, −7.62649384048912311495273401508, −6.35518029843768514515322988486, −5.95730287288048719998284934651, −4.87637136145488077840011961419, −4.30183055269319607962223826351, −3.34992733330930815714734321993, −2.55795775149010290881538005489, −1.94850892208441095473130446675, 0,
1.94850892208441095473130446675, 2.55795775149010290881538005489, 3.34992733330930815714734321993, 4.30183055269319607962223826351, 4.87637136145488077840011961419, 5.95730287288048719998284934651, 6.35518029843768514515322988486, 7.62649384048912311495273401508, 8.046869739368987952116016597995