L(s) = 1 | + 3-s − 2·5-s + 4·7-s + 9-s + 2·11-s + 6·13-s − 2·15-s − 6·19-s + 4·21-s − 25-s + 27-s + 6·29-s − 4·31-s + 2·33-s − 8·35-s − 2·37-s + 6·39-s + 6·41-s − 2·43-s − 2·45-s + 4·47-s + 9·49-s + 14·53-s − 4·55-s − 6·57-s − 4·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 0.516·15-s − 1.37·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.35·35-s − 0.328·37-s + 0.960·39-s + 0.937·41-s − 0.304·43-s − 0.298·45-s + 0.583·47-s + 9/7·49-s + 1.92·53-s − 0.539·55-s − 0.794·57-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.265169718\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.265169718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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
−16.08819604036670, −15.67914234096790, −15.09951936077731, −14.64557842272568, −14.12602708537238, −13.61155888949480, −12.92758300452340, −12.23513340388058, −11.64980932036043, −11.12569532105665, −10.76765950628994, −10.02924118819919, −8.964039069229445, −8.543254972697866, −8.309883914739435, −7.603022479050704, −6.945372148697959, −6.192303800089676, −5.432600951333801, −4.461035850676079, −4.071256937359413, −3.566057001939091, −2.415762663294877, −1.643818872988273, −0.8485740766815657,
0.8485740766815657, 1.643818872988273, 2.415762663294877, 3.566057001939091, 4.071256937359413, 4.461035850676079, 5.432600951333801, 6.192303800089676, 6.945372148697959, 7.603022479050704, 8.309883914739435, 8.543254972697866, 8.964039069229445, 10.02924118819919, 10.76765950628994, 11.12569532105665, 11.64980932036043, 12.23513340388058, 12.92758300452340, 13.61155888949480, 14.12602708537238, 14.64557842272568, 15.09951936077731, 15.67914234096790, 16.08819604036670