L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 2·11-s − 3·12-s + 16-s + 4·17-s − 6·18-s − 6·19-s + 2·22-s − 3·23-s + 3·24-s − 9·27-s + 9·29-s − 4·31-s − 32-s + 6·33-s − 4·34-s + 6·36-s + 4·37-s + 6·38-s − 7·41-s + 5·43-s − 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 0.603·11-s − 0.866·12-s + 1/4·16-s + 0.970·17-s − 1.41·18-s − 1.37·19-s + 0.426·22-s − 0.625·23-s + 0.612·24-s − 1.73·27-s + 1.67·29-s − 0.718·31-s − 0.176·32-s + 1.04·33-s − 0.685·34-s + 36-s + 0.657·37-s + 0.973·38-s − 1.09·41-s + 0.762·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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
−8.417499241675123169379683911165, −7.82144721582619846199525525154, −6.79142823413411428919178509257, −6.35867540140337573543266952016, −5.51539881626000827397282595258, −4.86653803922812333477500567244, −3.81411575748512984233099167890, −2.34942842423086253703244305252, −1.09953296358837609314631325079, 0,
1.09953296358837609314631325079, 2.34942842423086253703244305252, 3.81411575748512984233099167890, 4.86653803922812333477500567244, 5.51539881626000827397282595258, 6.35867540140337573543266952016, 6.79142823413411428919178509257, 7.82144721582619846199525525154, 8.417499241675123169379683911165