| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s − 6·11-s + 12-s + 5·13-s + 14-s + 16-s + 3·17-s + 2·18-s + 19-s − 21-s + 6·22-s + 3·23-s − 24-s − 5·25-s − 5·26-s − 5·27-s − 28-s + 9·29-s − 4·31-s − 32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 1.80·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s + 1.27·22-s + 0.625·23-s − 0.204·24-s − 25-s − 0.980·26-s − 0.962·27-s − 0.188·28-s + 1.67·29-s − 0.718·31-s − 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6302107433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6302107433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−16.22302156666961409355458160527, −15.51354522269081565751677762935, −14.02216922667742784390855478973, −12.90671167977020138547910461547, −11.24807629195878921565630077767, −10.09897479975324709020702728875, −8.684413720393242419277779815238, −7.73591843742988285704805588989, −5.81027585652647529822118508915, −2.99432139105097413830284496966,
2.99432139105097413830284496966, 5.81027585652647529822118508915, 7.73591843742988285704805588989, 8.684413720393242419277779815238, 10.09897479975324709020702728875, 11.24807629195878921565630077767, 12.90671167977020138547910461547, 14.02216922667742784390855478973, 15.51354522269081565751677762935, 16.22302156666961409355458160527
