| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 2·11-s − 6·13-s + 15-s + 2·19-s − 4·21-s + 25-s − 27-s + 6·29-s + 4·31-s + 2·33-s − 4·35-s + 4·37-s + 6·39-s − 2·41-s − 43-s − 45-s + 9·49-s + 2·55-s − 2·57-s + 10·59-s − 14·61-s + 4·63-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.258·15-s + 0.458·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s − 0.676·35-s + 0.657·37-s + 0.960·39-s − 0.312·41-s − 0.152·43-s − 0.149·45-s + 9/7·49-s + 0.269·55-s − 0.264·57-s + 1.30·59-s − 1.79·61-s + 0.503·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.07555209088466, −14.49279541052184, −14.10973928752772, −13.47591835225584, −12.82372483152154, −12.19017556497612, −11.82634068086118, −11.55875494242720, −10.79655005345274, −10.41344097389489, −9.885120832050582, −9.239608415011137, −8.427331043709122, −8.032471910312432, −7.496987951562873, −7.126900841198932, −6.357591046233413, −5.524667032655400, −5.115979216808264, −4.522071215818021, −4.324525714591570, −3.075219669875810, −2.547088593541670, −1.724557034975794, −0.9360245935090651, 0,
0.9360245935090651, 1.724557034975794, 2.547088593541670, 3.075219669875810, 4.324525714591570, 4.522071215818021, 5.115979216808264, 5.524667032655400, 6.357591046233413, 7.126900841198932, 7.496987951562873, 8.032471910312432, 8.427331043709122, 9.239608415011137, 9.885120832050582, 10.41344097389489, 10.79655005345274, 11.55875494242720, 11.82634068086118, 12.19017556497612, 12.82372483152154, 13.47591835225584, 14.10973928752772, 14.49279541052184, 15.07555209088466
