L(s) = 1 | − 5-s + 2·7-s + 4·11-s − 13-s − 4·17-s + 2·19-s + 2·23-s + 25-s − 8·29-s − 4·31-s − 2·35-s + 6·37-s − 10·41-s − 4·43-s − 3·49-s − 6·53-s − 4·55-s − 12·59-s − 2·61-s + 65-s + 8·67-s + 8·77-s + 8·79-s − 12·83-s + 4·85-s + 10·89-s − 2·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.20·11-s − 0.277·13-s − 0.970·17-s + 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.48·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 3/7·49-s − 0.824·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s + 0.124·65-s + 0.977·67-s + 0.911·77-s + 0.900·79-s − 1.31·83-s + 0.433·85-s + 1.05·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33084856839772916354379013452, −6.79381105169399316682525419831, −6.06647141156581861568898283442, −5.15454374478009007061603252008, −4.59258669700534138293277067762, −3.86220003031797799074822990754, −3.17601250555715120436069996640, −2.00718371760509793400260562245, −1.34608167335457443774095101092, 0,
1.34608167335457443774095101092, 2.00718371760509793400260562245, 3.17601250555715120436069996640, 3.86220003031797799074822990754, 4.59258669700534138293277067762, 5.15454374478009007061603252008, 6.06647141156581861568898283442, 6.79381105169399316682525419831, 7.33084856839772916354379013452