L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 2·11-s − 3·13-s − 14-s + 16-s − 17-s − 20-s − 2·22-s − 3·23-s + 25-s − 3·26-s − 28-s − 3·29-s − 31-s + 32-s − 34-s + 35-s − 4·37-s − 40-s − 10·41-s + 43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s − 0.588·26-s − 0.188·28-s − 0.557·29-s − 0.179·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s − 0.657·37-s − 0.158·40-s − 1.56·41-s + 0.152·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.727052397252795926230629180561, −7.893376560936596900147626105015, −7.17672791482183733005803579219, −6.44591022116589879583805475203, −5.44039838865113315898785990729, −4.77661706985340724100907932925, −3.80091664409660566693823190011, −2.97318127077761730237595821213, −1.92173560705050449098514074714, 0,
1.92173560705050449098514074714, 2.97318127077761730237595821213, 3.80091664409660566693823190011, 4.77661706985340724100907932925, 5.44039838865113315898785990729, 6.44591022116589879583805475203, 7.17672791482183733005803579219, 7.893376560936596900147626105015, 8.727052397252795926230629180561