L(s) = 1 | − 7-s + 2·11-s − 2·13-s + 6·19-s + 6·29-s − 10·31-s − 6·41-s + 8·43-s − 12·47-s + 49-s + 6·53-s − 6·61-s − 4·67-s + 6·71-s − 14·73-s − 2·77-s − 4·79-s + 6·89-s + 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.603·11-s − 0.554·13-s + 1.37·19-s + 1.11·29-s − 1.79·31-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s − 0.768·61-s − 0.488·67-s + 0.712·71-s − 1.63·73-s − 0.227·77-s − 0.450·79-s + 0.635·89-s + 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−15.82357795973992, −14.88562497118678, −14.67158273301336, −14.00916341510216, −13.53517894028912, −12.94607604550727, −12.30446127047958, −11.93616921397878, −11.36069168272423, −10.76059977478684, −10.02595024402157, −9.684914153160644, −9.045693494634208, −8.593142477277432, −7.695989256002508, −7.306843798266341, −6.702482191964183, −6.054637434562706, −5.383238001799175, −4.832308999841741, −4.047276290995982, −3.357613050934730, −2.816503941818100, −1.852824428251127, −1.079195378202583, 0,
1.079195378202583, 1.852824428251127, 2.816503941818100, 3.357613050934730, 4.047276290995982, 4.832308999841741, 5.383238001799175, 6.054637434562706, 6.702482191964183, 7.306843798266341, 7.695989256002508, 8.593142477277432, 9.045693494634208, 9.684914153160644, 10.02595024402157, 10.76059977478684, 11.36069168272423, 11.93616921397878, 12.30446127047958, 12.94607604550727, 13.53517894028912, 14.00916341510216, 14.67158273301336, 14.88562497118678, 15.82357795973992