L(s) = 1 | + 5-s − 7-s + 5·13-s − 6·17-s + 19-s − 4·23-s − 4·25-s + 29-s − 10·31-s − 35-s + 37-s + 8·43-s − 47-s + 49-s − 8·53-s + 3·59-s + 6·61-s + 5·65-s + 13·67-s + 3·73-s + 8·79-s + 2·83-s − 6·85-s + 6·89-s − 5·91-s + 95-s − 16·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.38·13-s − 1.45·17-s + 0.229·19-s − 0.834·23-s − 4/5·25-s + 0.185·29-s − 1.79·31-s − 0.169·35-s + 0.164·37-s + 1.21·43-s − 0.145·47-s + 1/7·49-s − 1.09·53-s + 0.390·59-s + 0.768·61-s + 0.620·65-s + 1.58·67-s + 0.351·73-s + 0.900·79-s + 0.219·83-s − 0.650·85-s + 0.635·89-s − 0.524·91-s + 0.102·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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
−14.34671098584056, −14.04137232802669, −13.49279870422066, −13.06979905353787, −12.71312445759362, −12.02531726426404, −11.37981152082640, −10.93285458130350, −10.67258448119149, −9.790214295320350, −9.451997101841992, −8.965012644728895, −8.374643849377530, −7.901560077833381, −7.148455508178662, −6.615622965559943, −6.101221148156102, −5.696750317513277, −5.046111266766255, −4.156528247210862, −3.838027288785030, −3.179240306727282, −2.202133541030622, −1.912596261004074, −0.9288242019688921, 0,
0.9288242019688921, 1.912596261004074, 2.202133541030622, 3.179240306727282, 3.838027288785030, 4.156528247210862, 5.046111266766255, 5.696750317513277, 6.101221148156102, 6.615622965559943, 7.148455508178662, 7.901560077833381, 8.374643849377530, 8.965012644728895, 9.451997101841992, 9.790214295320350, 10.67258448119149, 10.93285458130350, 11.37981152082640, 12.02531726426404, 12.71312445759362, 13.06979905353787, 13.49279870422066, 14.04137232802669, 14.34671098584056