L(s) = 1 | − 0.494·2-s + 3-s − 1.75·4-s − 0.494·6-s + 1.85·8-s + 9-s + 5.33·11-s − 1.75·12-s − 5.09·13-s + 2.59·16-s − 0.350·17-s − 0.494·18-s − 2.76·19-s − 2.63·22-s + 7.51·23-s + 1.85·24-s + 2.51·26-s + 27-s + 4·29-s − 6.11·31-s − 4.99·32-s + 5.33·33-s + 0.173·34-s − 1.75·36-s − 3.52·37-s + 1.36·38-s − 5.09·39-s + ⋯ |
L(s) = 1 | − 0.349·2-s + 0.577·3-s − 0.877·4-s − 0.201·6-s + 0.656·8-s + 0.333·9-s + 1.60·11-s − 0.506·12-s − 1.41·13-s + 0.648·16-s − 0.0849·17-s − 0.116·18-s − 0.634·19-s − 0.562·22-s + 1.56·23-s + 0.378·24-s + 0.493·26-s + 0.192·27-s + 0.742·29-s − 1.09·31-s − 0.882·32-s + 0.929·33-s + 0.0296·34-s − 0.292·36-s − 0.579·37-s + 0.221·38-s − 0.815·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.597145050\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597145050\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 0.494T + 2T^{2} \) |
| 11 | \( 1 - 5.33T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.350T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 7.51T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 - 1.41T + 43T^{2} \) |
| 47 | \( 1 - 8.85T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 3.75T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 - 2.36T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−8.777471543684285206632247299590, −7.86720313320117827192698994070, −7.19586534282051730550027699339, −6.52879203545871027914901475814, −5.35969805260832104299437542366, −4.57458359610688997249813230724, −3.98970806971868535235730001616, −3.03754690436243456373126155650, −1.88962634918161138770559425348, −0.78593594889558566311843536902,
0.78593594889558566311843536902, 1.88962634918161138770559425348, 3.03754690436243456373126155650, 3.98970806971868535235730001616, 4.57458359610688997249813230724, 5.35969805260832104299437542366, 6.52879203545871027914901475814, 7.19586534282051730550027699339, 7.86720313320117827192698994070, 8.777471543684285206632247299590