L(s) = 1 | − 2-s + 1.76·3-s + 4-s − 4.09·5-s − 1.76·6-s + 4.14·7-s − 8-s + 0.112·9-s + 4.09·10-s − 5.57·11-s + 1.76·12-s + 2.12·13-s − 4.14·14-s − 7.21·15-s + 16-s + 3.96·17-s − 0.112·18-s − 8.26·19-s − 4.09·20-s + 7.32·21-s + 5.57·22-s + 6.53·23-s − 1.76·24-s + 11.7·25-s − 2.12·26-s − 5.09·27-s + 4.14·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.01·3-s + 0.5·4-s − 1.82·5-s − 0.720·6-s + 1.56·7-s − 0.353·8-s + 0.0375·9-s + 1.29·10-s − 1.68·11-s + 0.509·12-s + 0.590·13-s − 1.10·14-s − 1.86·15-s + 0.250·16-s + 0.961·17-s − 0.0265·18-s − 1.89·19-s − 0.914·20-s + 1.59·21-s + 1.18·22-s + 1.36·23-s − 0.360·24-s + 2.34·25-s − 0.417·26-s − 0.980·27-s + 0.784·28-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 | \( 1 - 1.76T + 3T^{2} \) |
| 5 | \( 1 + 4.09T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 8.26T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 + 0.348T + 29T^{2} \) |
| 31 | \( 1 - 5.91T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.45T + 67T^{2} \) |
| 71 | \( 1 + 4.62T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 + 1.84T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−8.163646217118578614980857443831, −7.69517363339638909387083756394, −7.24992519202062504509305658594, −5.88472864934695398679788263654, −4.79968413792106251028854336945, −4.25881856706768784724345865667, −3.14859973698603925431763731460, −2.61285535633622267744538696272, −1.35832588137743795112430323026, 0,
1.35832588137743795112430323026, 2.61285535633622267744538696272, 3.14859973698603925431763731460, 4.25881856706768784724345865667, 4.79968413792106251028854336945, 5.88472864934695398679788263654, 7.24992519202062504509305658594, 7.69517363339638909387083756394, 8.163646217118578614980857443831