| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 17-s − 4·19-s − 25-s − 27-s + 10·29-s + 8·31-s + 4·33-s + 2·37-s + 2·39-s − 10·41-s + 12·43-s − 2·45-s + 51-s − 6·53-s + 8·55-s + 4·57-s − 12·59-s − 10·61-s + 4·65-s − 12·67-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.242·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s + 0.140·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 0.496·65-s − 1.46·67-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.65156411202127, −12.92396987826918, −12.35180301934027, −12.16852202664667, −11.73068207750183, −11.10528057420719, −10.60852387650484, −10.36786217109513, −9.874225405181623, −9.167296894049193, −8.613537644464530, −8.047185306958612, −7.783139736765610, −7.271597204331422, −6.608855115208004, −6.192132031593576, −5.689558572361227, −4.862613714192066, −4.540454090955972, −4.320809369214716, −3.295744818960005, −2.866374919973827, −2.286563819458015, −1.448100008135909, −0.5680019784979377, 0,
0.5680019784979377, 1.448100008135909, 2.286563819458015, 2.866374919973827, 3.295744818960005, 4.320809369214716, 4.540454090955972, 4.862613714192066, 5.689558572361227, 6.192132031593576, 6.608855115208004, 7.271597204331422, 7.783139736765610, 8.047185306958612, 8.613537644464530, 9.167296894049193, 9.874225405181623, 10.36786217109513, 10.60852387650484, 11.10528057420719, 11.73068207750183, 12.16852202664667, 12.35180301934027, 12.92396987826918, 13.65156411202127
