| L(s) = 1 | − 3-s + 2·5-s + 9-s − 4·11-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 − 10·41-s − 12·43-s + 2·45-s − 7·49-s − 51-s + 6·53-s − 8·55-s − 4·57-s + 12·59-s − 10·61-s − 12·67-s − 10·73-s + 75-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-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 − 1.56·41-s − 1.82·43-s + 0.298·45-s − 49-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 − 1.46·67-s − 1.17·73-s + 0.115·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.50420444461656, −13.19234232918619, −12.96713566244213, −12.06570448346607, −11.73099134611472, −11.42362052132259, −10.57575360113041, −10.33678068627147, −9.851578636774246, −9.547524191766623, −8.842149518861327, −8.268627740680029, −7.704620266220823, −7.335799404366824, −6.641391895521599, −6.164915788164104, −5.625607954698156, −5.221272904483018, −4.872624564276494, −4.122479546579892, −3.259254264475711, −2.981060025434821, −1.945895556502183, −1.771343207289496, −0.7784420356271485, 0,
0.7784420356271485, 1.771343207289496, 1.945895556502183, 2.981060025434821, 3.259254264475711, 4.122479546579892, 4.872624564276494, 5.221272904483018, 5.625607954698156, 6.164915788164104, 6.641391895521599, 7.335799404366824, 7.704620266220823, 8.268627740680029, 8.842149518861327, 9.547524191766623, 9.851578636774246, 10.33678068627147, 10.57575360113041, 11.42362052132259, 11.73099134611472, 12.06570448346607, 12.96713566244213, 13.19234232918619, 13.50420444461656
