L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 3·11-s − 5·13-s − 14-s + 16-s − 4·17-s + 19-s − 3·22-s − 8·23-s + 5·26-s + 28-s + 10·29-s − 8·31-s − 32-s + 4·34-s + 4·37-s − 38-s + 9·41-s − 43-s + 3·44-s + 8·46-s + 13·47-s + 49-s − 5·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.904·11-s − 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.229·19-s − 0.639·22-s − 1.66·23-s + 0.980·26-s + 0.188·28-s + 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.685·34-s + 0.657·37-s − 0.162·38-s + 1.40·41-s − 0.152·43-s + 0.452·44-s + 1.17·46-s + 1.89·47-s + 1/7·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.42586797528659619841276099281, −6.82680331542778916881461208316, −6.13869319740432186719851855220, −5.40145993911283207807247423970, −4.42864724827216082742667479435, −3.98348454991716067345430916932, −2.68604080369804976506090869073, −2.17512229159840888619864961267, −1.15944074715689730465091068288, 0,
1.15944074715689730465091068288, 2.17512229159840888619864961267, 2.68604080369804976506090869073, 3.98348454991716067345430916932, 4.42864724827216082742667479435, 5.40145993911283207807247423970, 6.13869319740432186719851855220, 6.82680331542778916881461208316, 7.42586797528659619841276099281