L(s) = 1 | − 7-s − 6·11-s + 2·13-s + 4·19-s − 6·23-s − 5·25-s − 6·29-s − 8·31-s + 2·37-s − 12·41-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 10·61-s − 8·67-s + 6·71-s − 10·73-s + 6·77-s + 4·79-s − 12·83-s − 12·89-s − 2·91-s − 10·97-s + 12·101-s − 8·103-s − 6·107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.917·19-s − 1.25·23-s − 25-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s − 0.977·67-s + 0.712·71-s − 1.17·73-s + 0.683·77-s + 0.450·79-s − 1.31·83-s − 1.27·89-s − 0.209·91-s − 1.01·97-s + 1.19·101-s − 0.788·103-s − 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−9.666722228038389417268306810761, −8.701525582792892098188512767049, −7.76051152200915202719683008376, −7.26031081711332154770334058409, −5.82277620068151436147210793072, −5.48036643270637995428248023009, −4.11399937114205741213147454042, −3.12875372570045370376261059932, −1.96257743518867342978553596046, 0,
1.96257743518867342978553596046, 3.12875372570045370376261059932, 4.11399937114205741213147454042, 5.48036643270637995428248023009, 5.82277620068151436147210793072, 7.26031081711332154770334058409, 7.76051152200915202719683008376, 8.701525582792892098188512767049, 9.666722228038389417268306810761