L(s) = 1 | + 5-s + 7-s − 3·9-s + 11-s + 2·13-s + 6·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s + 35-s − 10·37-s − 6·41-s − 12·43-s − 3·45-s − 12·47-s + 49-s + 6·53-s + 55-s + 12·59-s + 6·61-s − 3·63-s + 2·65-s − 8·67-s + 8·71-s + 14·73-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s + 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s − 1.82·43-s − 0.447·45-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 1.56·59-s + 0.768·61-s − 0.377·63-s + 0.248·65-s − 0.977·67-s + 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.86753159587685737017972837753, −6.87560183309621907612322969680, −6.28826610755447419210134087660, −5.39056864125626154103768346357, −5.18346287795805451952417926396, −3.77325544323858382218177542007, −3.39719426683704149293632831037, −2.17216060445891897152140084292, −1.47543790752666454565017904453, 0,
1.47543790752666454565017904453, 2.17216060445891897152140084292, 3.39719426683704149293632831037, 3.77325544323858382218177542007, 5.18346287795805451952417926396, 5.39056864125626154103768346357, 6.28826610755447419210134087660, 6.87560183309621907612322969680, 7.86753159587685737017972837753