| L(s) = 1 | − 2-s + 3-s + 4-s + 1.27·5-s − 6-s − 5.23·7-s − 8-s + 9-s − 1.27·10-s − 0.0462·11-s + 12-s − 5.72·13-s + 5.23·14-s + 1.27·15-s + 16-s + 5.23·17-s − 18-s + 8.21·19-s + 1.27·20-s − 5.23·21-s + 0.0462·22-s + 23-s − 24-s − 3.37·25-s + 5.72·26-s + 27-s − 5.23·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.570·5-s − 0.408·6-s − 1.97·7-s − 0.353·8-s + 0.333·9-s − 0.403·10-s − 0.0139·11-s + 0.288·12-s − 1.58·13-s + 1.39·14-s + 0.329·15-s + 0.250·16-s + 1.26·17-s − 0.235·18-s + 1.88·19-s + 0.285·20-s − 1.14·21-s + 0.00986·22-s + 0.208·23-s − 0.204·24-s − 0.674·25-s + 1.12·26-s + 0.192·27-s − 0.988·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 + 5.23T + 7T^{2} \) |
| 11 | \( 1 + 0.0462T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 8.21T + 19T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 7.63T + 59T^{2} \) |
| 61 | \( 1 + 5.95T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.006201697023749982699187873296, −7.28854899532526035128345655810, −6.97451200415505683819742099395, −5.84205840310264702611432757761, −5.43840978305669829066686370821, −3.98097171814116112165501918503, −2.94163769218434270845997362482, −2.77138212184333404659683100256, −1.35396016070446343617098042527, 0,
1.35396016070446343617098042527, 2.77138212184333404659683100256, 2.94163769218434270845997362482, 3.98097171814116112165501918503, 5.43840978305669829066686370821, 5.84205840310264702611432757761, 6.97451200415505683819742099395, 7.28854899532526035128345655810, 8.006201697023749982699187873296
