| L(s) = 1 | − 1.05·2-s − 3-s − 0.877·4-s − 1.30·5-s + 1.05·6-s + 0.720·7-s + 3.04·8-s + 9-s + 1.38·10-s + 4.26·11-s + 0.877·12-s + 6.49·13-s − 0.763·14-s + 1.30·15-s − 1.47·16-s + 1.60·17-s − 1.05·18-s + 0.0676·19-s + 1.14·20-s − 0.720·21-s − 4.52·22-s + 23-s − 3.04·24-s − 3.29·25-s − 6.88·26-s − 27-s − 0.632·28-s + ⋯ |
| L(s) = 1 | − 0.749·2-s − 0.577·3-s − 0.438·4-s − 0.583·5-s + 0.432·6-s + 0.272·7-s + 1.07·8-s + 0.333·9-s + 0.436·10-s + 1.28·11-s + 0.253·12-s + 1.80·13-s − 0.204·14-s + 0.336·15-s − 0.369·16-s + 0.389·17-s − 0.249·18-s + 0.0155·19-s + 0.255·20-s − 0.157·21-s − 0.964·22-s + 0.208·23-s − 0.622·24-s − 0.659·25-s − 1.34·26-s − 0.192·27-s − 0.119·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8868038377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8868038377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 0.720T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 6.49T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 0.0676T + 19T^{2} \) |
| 31 | \( 1 + 3.80T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 + 0.467T + 41T^{2} \) |
| 43 | \( 1 - 0.971T + 43T^{2} \) |
| 47 | \( 1 - 7.97T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 7.29T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 0.843T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 8.90T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 0.847T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.997655152643413804205740916514, −8.555061597784624991304087895347, −7.72864532084800450634001003620, −6.93344983052989983755418523429, −6.03986850268646343470434998266, −5.18054508746570326451030129709, −3.96317066136657062670164950797, −3.77462747492183955303919749916, −1.64260965043072247554890306517, −0.799474341492482698311904334842,
0.799474341492482698311904334842, 1.64260965043072247554890306517, 3.77462747492183955303919749916, 3.96317066136657062670164950797, 5.18054508746570326451030129709, 6.03986850268646343470434998266, 6.93344983052989983755418523429, 7.72864532084800450634001003620, 8.555061597784624991304087895347, 8.997655152643413804205740916514
