L(s) = 1 | − 8·4-s − 31.1·7-s + 64·16-s + 155.·19-s − 125·25-s + 249.·28-s − 155.·31-s + 436.·37-s + 520·43-s + 629·49-s − 182·61-s − 512·64-s − 654.·67-s + 374.·73-s − 1.24e3·76-s − 884·79-s − 1.37e3·97-s + 1.00e3·100-s − 1.82e3·103-s + 2.18e3·109-s − 1.99e3·112-s + ⋯ |
L(s) = 1 | − 4-s − 1.68·7-s + 16-s + 1.88·19-s − 25-s + 1.68·28-s − 0.903·31-s + 1.93·37-s + 1.84·43-s + 1.83·49-s − 0.382·61-s − 64-s − 1.19·67-s + 0.599·73-s − 1.88·76-s − 1.25·79-s − 1.43·97-s + 100-s − 1.74·103-s + 1.91·109-s − 1.68·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 + 31.1T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 + 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 436.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 182T + 2.26e5T^{2} \) |
| 67 | \( 1 + 654.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 - 374.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 884T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3T + 9.12e5T^{2} \) |
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
−8.998214003713661364437147075987, −7.83327389115685696044828067506, −7.21079640178592173931393431180, −6.01113100590954679548542440306, −5.57895385905292561629522994632, −4.33717937869256377580484571026, −3.55505660731407155280105406811, −2.77187807113650451896034189970, −1.00655724279934310474327491070, 0,
1.00655724279934310474327491070, 2.77187807113650451896034189970, 3.55505660731407155280105406811, 4.33717937869256377580484571026, 5.57895385905292561629522994632, 6.01113100590954679548542440306, 7.21079640178592173931393431180, 7.83327389115685696044828067506, 8.998214003713661364437147075987