L(s) = 1 | − 3·3-s − 2·4-s − 5-s − 7-s + 6·9-s − 11-s + 6·12-s − 4·13-s + 3·15-s + 4·16-s + 2·17-s − 6·19-s + 2·20-s + 3·21-s − 5·23-s − 4·25-s − 9·27-s + 2·28-s + 10·29-s + 31-s + 3·33-s + 35-s − 12·36-s − 5·37-s + 12·39-s − 2·41-s − 8·43-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 4-s − 0.447·5-s − 0.377·7-s + 2·9-s − 0.301·11-s + 1.73·12-s − 1.10·13-s + 0.774·15-s + 16-s + 0.485·17-s − 1.37·19-s + 0.447·20-s + 0.654·21-s − 1.04·23-s − 4/5·25-s − 1.73·27-s + 0.377·28-s + 1.85·29-s + 0.179·31-s + 0.522·33-s + 0.169·35-s − 2·36-s − 0.821·37-s + 1.92·39-s − 0.312·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.75798057830490060611386820011, −12.41538070383590792560753216926, −12.09579448828259015201597448613, −10.54462273276049844454718422984, −9.842751379499007423618867754387, −8.090125574816450530034167252480, −6.56709274805536909588701134945, −5.30170245060560361736841457140, −4.26020787258843561459488122037, 0,
4.26020787258843561459488122037, 5.30170245060560361736841457140, 6.56709274805536909588701134945, 8.090125574816450530034167252480, 9.842751379499007423618867754387, 10.54462273276049844454718422984, 12.09579448828259015201597448613, 12.41538070383590792560753216926, 13.75798057830490060611386820011