| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s + 4·13-s + 16-s + 4·17-s − 18-s − 19-s − 22-s + 5·23-s − 24-s − 4·26-s + 27-s + 3·29-s − 5·31-s − 32-s + 33-s − 4·34-s + 36-s − 6·37-s + 38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.229·19-s − 0.213·22-s + 1.04·23-s − 0.204·24-s − 0.784·26-s + 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.176·32-s + 0.174·33-s − 0.685·34-s + 1/6·36-s − 0.986·37-s + 0.162·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864234236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864234236\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.882020288898595720205483943624, −8.136907463287869713982480423542, −7.44168296487690896909503280077, −6.67990441300417068394374969660, −5.89827362371595823921949426579, −4.91316767246911379411451363673, −3.71067462179144069924878338098, −3.13067692404062497463207516829, −1.90860784806972172503187361888, −0.966041724424292256577588335974,
0.966041724424292256577588335974, 1.90860784806972172503187361888, 3.13067692404062497463207516829, 3.71067462179144069924878338098, 4.91316767246911379411451363673, 5.89827362371595823921949426579, 6.67990441300417068394374969660, 7.44168296487690896909503280077, 8.136907463287869713982480423542, 8.882020288898595720205483943624
