L(s) = 1 | − 1.31·2-s − 3-s − 0.260·4-s + 2.51·5-s + 1.31·6-s − 2.25·7-s + 2.98·8-s + 9-s − 3.31·10-s − 0.206·11-s + 0.260·12-s − 0.720·13-s + 2.97·14-s − 2.51·15-s − 3.41·16-s + 2.26·17-s − 1.31·18-s − 4.58·19-s − 0.655·20-s + 2.25·21-s + 0.272·22-s + 23-s − 2.98·24-s + 1.31·25-s + 0.950·26-s − 27-s + 0.587·28-s + ⋯ |
L(s) = 1 | − 0.932·2-s − 0.577·3-s − 0.130·4-s + 1.12·5-s + 0.538·6-s − 0.851·7-s + 1.05·8-s + 0.333·9-s − 1.04·10-s − 0.0623·11-s + 0.0753·12-s − 0.199·13-s + 0.793·14-s − 0.649·15-s − 0.852·16-s + 0.548·17-s − 0.310·18-s − 1.05·19-s − 0.146·20-s + 0.491·21-s + 0.0581·22-s + 0.208·23-s − 0.608·24-s + 0.263·25-s + 0.186·26-s − 0.192·27-s + 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 1.31T + 2T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 + 2.25T + 7T^{2} \) |
| 11 | \( 1 + 0.206T + 11T^{2} \) |
| 13 | \( 1 + 0.720T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 31 | \( 1 + 0.615T + 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 - 5.61T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 - 9.36T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 - 1.72T + 97T^{2} \) |
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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
−8.947905781639100994904443537068, −8.178327029917462077016428818851, −7.14694364453241280357832487862, −6.47227628374609674729135585437, −5.67937985869290355351448833926, −4.88564914494715297849902505642, −3.80879045957945628773915433500, −2.43042810073001041573317296685, −1.34312554582318347956188047667, 0,
1.34312554582318347956188047667, 2.43042810073001041573317296685, 3.80879045957945628773915433500, 4.88564914494715297849902505642, 5.67937985869290355351448833926, 6.47227628374609674729135585437, 7.14694364453241280357832487862, 8.178327029917462077016428818851, 8.947905781639100994904443537068