L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 13-s + 15-s + 4·17-s − 2·19-s − 2·21-s − 6·23-s + 25-s + 27-s − 4·31-s − 2·35-s − 2·37-s − 39-s − 6·41-s − 4·43-s + 45-s − 4·47-s − 3·49-s + 4·51-s − 10·53-s − 2·57-s − 8·59-s + 6·61-s − 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.970·17-s − 0.458·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.560·51-s − 1.37·53-s − 0.264·57-s − 1.04·59-s + 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.926155056817298276567326919502, −6.86581770911850411619241659882, −6.42264684442568383299513179361, −5.57162117864929809752652502616, −4.85022187973712501953460698095, −3.77919191674874488252110367603, −3.27725268260994039389931177496, −2.32305097962805257877859405728, −1.51252849433525315148657124925, 0,
1.51252849433525315148657124925, 2.32305097962805257877859405728, 3.27725268260994039389931177496, 3.77919191674874488252110367603, 4.85022187973712501953460698095, 5.57162117864929809752652502616, 6.42264684442568383299513179361, 6.86581770911850411619241659882, 7.926155056817298276567326919502