L(s) = 1 | − 3·7-s + 11-s − 13-s + 5·17-s + 4·19-s + 2·23-s − 3·29-s + 5·31-s + 6·37-s + 8·41-s + 6·43-s + 47-s + 2·49-s + 11·53-s + 3·59-s + 5·61-s − 9·67-s − 8·71-s − 2·73-s − 3·77-s − 4·79-s − 3·83-s − 6·89-s + 3·91-s − 4·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 0.301·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s + 0.417·23-s − 0.557·29-s + 0.898·31-s + 0.986·37-s + 1.24·41-s + 0.914·43-s + 0.145·47-s + 2/7·49-s + 1.51·53-s + 0.390·59-s + 0.640·61-s − 1.09·67-s − 0.949·71-s − 0.234·73-s − 0.341·77-s − 0.450·79-s − 0.329·83-s − 0.635·89-s + 0.314·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.24330043128274, −12.95795995732872, −12.37860481683506, −12.01986904052840, −11.57677017906199, −11.04076863298028, −10.38237068128141, −10.03406563770711, −9.499203316667013, −9.313749255424040, −8.677597744121927, −8.030839210345472, −7.494079094706936, −7.198625279363077, −6.582030813629385, −6.030664039217665, −5.598532404764111, −5.201988259229555, −4.239111355797363, −4.036512316914141, −3.226149645613571, −2.852991043143335, −2.373108058374255, −1.246113329654720, −0.9333477084455270, 0,
0.9333477084455270, 1.246113329654720, 2.373108058374255, 2.852991043143335, 3.226149645613571, 4.036512316914141, 4.239111355797363, 5.201988259229555, 5.598532404764111, 6.030664039217665, 6.582030813629385, 7.198625279363077, 7.494079094706936, 8.030839210345472, 8.677597744121927, 9.313749255424040, 9.499203316667013, 10.03406563770711, 10.38237068128141, 11.04076863298028, 11.57677017906199, 12.01986904052840, 12.37860481683506, 12.95795995732872, 13.24330043128274