| L(s) = 1 | − 1.05·2-s + 3-s − 0.891·4-s + 4.03·5-s − 1.05·6-s + 7-s + 3.04·8-s + 9-s − 4.24·10-s + 0.242·11-s − 0.891·12-s + 0.703·13-s − 1.05·14-s + 4.03·15-s − 1.42·16-s + 4.01·17-s − 1.05·18-s + 4.25·19-s − 3.59·20-s + 21-s − 0.254·22-s − 2.98·23-s + 3.04·24-s + 11.2·25-s − 0.741·26-s + 27-s − 0.891·28-s + ⋯ |
| L(s) = 1 | − 0.744·2-s + 0.577·3-s − 0.445·4-s + 1.80·5-s − 0.429·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s − 1.34·10-s + 0.0729·11-s − 0.257·12-s + 0.195·13-s − 0.281·14-s + 1.04·15-s − 0.355·16-s + 0.973·17-s − 0.248·18-s + 0.975·19-s − 0.804·20-s + 0.218·21-s − 0.0543·22-s − 0.623·23-s + 0.621·24-s + 2.25·25-s − 0.145·26-s + 0.192·27-s − 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.406147404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406147404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 - 4.03T + 5T^{2} \) |
| 11 | \( 1 - 0.242T + 11T^{2} \) |
| 13 | \( 1 - 0.703T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 + 1.99T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 0.658T + 67T^{2} \) |
| 71 | \( 1 + 0.254T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 1.66T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−8.374668596037138931675680017512, −8.150061952251193564496102928261, −7.05654853018006310544018891808, −6.34371798906953719040872284555, −5.29681792253186651812137059678, −4.97937875254695809884161366562, −3.72363689024879003500432716673, −2.72097128018588828060448532136, −1.70106196149592597380808597378, −1.10895965001790850036010105275,
1.10895965001790850036010105275, 1.70106196149592597380808597378, 2.72097128018588828060448532136, 3.72363689024879003500432716673, 4.97937875254695809884161366562, 5.29681792253186651812137059678, 6.34371798906953719040872284555, 7.05654853018006310544018891808, 8.150061952251193564496102928261, 8.374668596037138931675680017512
