| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 2·7-s + 3·8-s + 9-s + 2·12-s − 13-s − 2·14-s − 16-s + 5·17-s − 18-s + 6·19-s − 4·21-s − 2·23-s − 6·24-s + 26-s + 4·27-s − 2·28-s + 9·29-s − 2·31-s − 5·32-s − 5·34-s − 36-s + 3·37-s − 6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.577·12-s − 0.277·13-s − 0.534·14-s − 1/4·16-s + 1.21·17-s − 0.235·18-s + 1.37·19-s − 0.872·21-s − 0.417·23-s − 1.22·24-s + 0.196·26-s + 0.769·27-s − 0.377·28-s + 1.67·29-s − 0.359·31-s − 0.883·32-s − 0.857·34-s − 1/6·36-s + 0.493·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7451280270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7451280270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| 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
−8.579461901315169739376678660673, −8.069986483292601761137519952233, −7.37226685783266137740275636717, −6.48190836691386467410771774372, −5.38786861929707559822010222316, −5.14830451796636083715635540878, −4.26835168421269434433264314646, −3.07367978379982844766979708515, −1.49712391443289391355106255202, −0.68832603438404776244692791710,
0.68832603438404776244692791710, 1.49712391443289391355106255202, 3.07367978379982844766979708515, 4.26835168421269434433264314646, 5.14830451796636083715635540878, 5.38786861929707559822010222316, 6.48190836691386467410771774372, 7.37226685783266137740275636717, 8.069986483292601761137519952233, 8.579461901315169739376678660673
