L(s) = 1 | − 3·5-s + 7-s + 3·11-s − 4·13-s + 3·17-s − 19-s + 4·25-s − 6·29-s + 4·31-s − 3·35-s + 2·37-s + 6·41-s + 43-s − 3·47-s − 6·49-s − 12·53-s − 9·55-s − 6·59-s − 61-s + 12·65-s + 4·67-s + 6·71-s − 7·73-s + 3·77-s − 8·79-s + 12·83-s − 9·85-s + ⋯ |
L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.229·19-s + 4/5·25-s − 1.11·29-s + 0.718·31-s − 0.507·35-s + 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.437·47-s − 6/7·49-s − 1.64·53-s − 1.21·55-s − 0.781·59-s − 0.128·61-s + 1.48·65-s + 0.488·67-s + 0.712·71-s − 0.819·73-s + 0.341·77-s − 0.900·79-s + 1.31·83-s − 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−8.169758757825737858267666465656, −7.80441727232688730491090849584, −7.09896358802950320449173631503, −6.24695455384273016222733694355, −5.16165536636701751832135979028, −4.38988833936187387176065759569, −3.73798930153556479954881225054, −2.76577588204614961513690244188, −1.40889938157129298285707376171, 0,
1.40889938157129298285707376171, 2.76577588204614961513690244188, 3.73798930153556479954881225054, 4.38988833936187387176065759569, 5.16165536636701751832135979028, 6.24695455384273016222733694355, 7.09896358802950320449173631503, 7.80441727232688730491090849584, 8.169758757825737858267666465656