L(s) = 1 | − 2·2-s + 2·3-s + 4·4-s − 4·6-s − 8·8-s − 23·9-s − 28·11-s + 8·12-s + 12·13-s + 16·16-s − 64·17-s + 46·18-s + 60·19-s + 56·22-s + 58·23-s − 16·24-s − 24·26-s − 100·27-s + 90·29-s + 128·31-s − 32·32-s − 56·33-s + 128·34-s − 92·36-s − 236·37-s − 120·38-s + 24·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.272·6-s − 0.353·8-s − 0.851·9-s − 0.767·11-s + 0.192·12-s + 0.256·13-s + 1/4·16-s − 0.913·17-s + 0.602·18-s + 0.724·19-s + 0.542·22-s + 0.525·23-s − 0.136·24-s − 0.181·26-s − 0.712·27-s + 0.576·29-s + 0.741·31-s − 0.176·32-s − 0.295·33-s + 0.645·34-s − 0.425·36-s − 1.04·37-s − 0.512·38-s + 0.0985·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.113319948\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113319948\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 64 T + p^{3} T^{2} \) |
| 19 | \( 1 - 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 128 T + p^{3} T^{2} \) |
| 37 | \( 1 + 236 T + p^{3} T^{2} \) |
| 41 | \( 1 + 242 T + p^{3} T^{2} \) |
| 43 | \( 1 + 362 T + p^{3} T^{2} \) |
| 47 | \( 1 - 226 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 + 542 T + p^{3} T^{2} \) |
| 67 | \( 1 - 434 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 - 632 T + p^{3} T^{2} \) |
| 79 | \( 1 + 720 T + p^{3} T^{2} \) |
| 83 | \( 1 + 478 T + p^{3} T^{2} \) |
| 89 | \( 1 - 490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1456 T + p^{3} T^{2} \) |
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
−8.645726804087319293279850508035, −8.023816961345608917900614121835, −7.20543530056415563981151174857, −6.43084327006523472905530780945, −5.53158159446791779322289813210, −4.72084670890444112860475186871, −3.38184150303064431575487857062, −2.75326217726992453672745935728, −1.77875137616764951790212959227, −0.49273099076455869475803941641,
0.49273099076455869475803941641, 1.77875137616764951790212959227, 2.75326217726992453672745935728, 3.38184150303064431575487857062, 4.72084670890444112860475186871, 5.53158159446791779322289813210, 6.43084327006523472905530780945, 7.20543530056415563981151174857, 8.023816961345608917900614121835, 8.645726804087319293279850508035