| L(s) = 1 | + 2-s − 3-s + 4-s − 0.652·5-s − 6-s − 3.89·7-s + 8-s + 9-s − 0.652·10-s + 5.43·11-s − 12-s + 3.14·13-s − 3.89·14-s + 0.652·15-s + 16-s + 4.77·17-s + 18-s − 7.24·19-s − 0.652·20-s + 3.89·21-s + 5.43·22-s − 23-s − 24-s − 4.57·25-s + 3.14·26-s − 27-s − 3.89·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.292·5-s − 0.408·6-s − 1.47·7-s + 0.353·8-s + 0.333·9-s − 0.206·10-s + 1.63·11-s − 0.288·12-s + 0.871·13-s − 1.04·14-s + 0.168·15-s + 0.250·16-s + 1.15·17-s + 0.235·18-s − 1.66·19-s − 0.146·20-s + 0.850·21-s + 1.15·22-s − 0.208·23-s − 0.204·24-s − 0.914·25-s + 0.616·26-s − 0.192·27-s − 0.736·28-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
| \(L(1)\) |
\(\approx\) |
\(2.106926464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106926464\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.652T + 5T^{2} \) |
| 7 | \( 1 + 3.89T + 7T^{2} \) |
| 11 | \( 1 - 5.43T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + 1.28T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 - 9.97T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 + 4.99T + 67T^{2} \) |
| 71 | \( 1 + 0.305T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 8.46T + 83T^{2} \) |
| 89 | \( 1 - 7.44T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−8.514145209929140007297474725741, −7.43058332928863645183539403113, −6.61725892538844719689741717267, −6.24230625122690068411531527643, −5.74479431948903609799879438951, −4.50075422738214119321475005849, −3.75860635541933623105623822082, −3.41908874728328934385418670915, −2.01520868188976670209533314902, −0.77546503125119835742745990205,
0.77546503125119835742745990205, 2.01520868188976670209533314902, 3.41908874728328934385418670915, 3.75860635541933623105623822082, 4.50075422738214119321475005849, 5.74479431948903609799879438951, 6.24230625122690068411531527643, 6.61725892538844719689741717267, 7.43058332928863645183539403113, 8.514145209929140007297474725741
