L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s + 3·11-s + 12-s − 2·14-s + 15-s + 16-s + 6·17-s + 18-s − 2·19-s + 20-s − 2·21-s + 3·22-s + 3·23-s + 24-s + 25-s + 27-s − 2·28-s + 3·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.436·21-s + 0.639·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.486537778\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.486537778\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.184754709354177899704450853159, −7.37747572597708577983470069086, −6.67155537427429361138114387929, −6.11211364639362039757606172563, −5.32828772006042741623971114125, −4.45184752141749789969866997554, −3.54766044449277081308283996925, −3.10748682397540864644309548980, −2.08356120319711071770295965241, −1.07428484583357266222754713581,
1.07428484583357266222754713581, 2.08356120319711071770295965241, 3.10748682397540864644309548980, 3.54766044449277081308283996925, 4.45184752141749789969866997554, 5.32828772006042741623971114125, 6.11211364639362039757606172563, 6.67155537427429361138114387929, 7.37747572597708577983470069086, 8.184754709354177899704450853159