| L(s) = 1 | − 2·4-s + 5-s − 4·7-s − 3·9-s − 3·11-s + 2·13-s + 4·16-s − 4·17-s + 2·19-s − 2·20-s − 23-s − 4·25-s + 8·28-s − 2·29-s − 8·31-s − 4·35-s + 6·36-s + 7·37-s + 11·41-s + 3·43-s + 6·44-s − 3·45-s − 9·47-s + 9·49-s − 4·52-s + 9·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s − 9-s − 0.904·11-s + 0.554·13-s + 16-s − 0.970·17-s + 0.458·19-s − 0.447·20-s − 0.208·23-s − 4/5·25-s + 1.51·28-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 36-s + 1.15·37-s + 1.71·41-s + 0.457·43-s + 0.904·44-s − 0.447·45-s − 1.31·47-s + 9/7·49-s − 0.554·52-s + 1.23·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 269 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 269 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−11.40503845862669404741197697105, −10.31767650394093071905287023450, −9.429748042845105458622615719434, −8.851224139712706448083893835624, −7.63939050545091637195351002555, −6.13065631909578093325834919161, −5.52035129648516313797764764737, −3.96033722802915316882524059193, −2.76936071771683709295115864629, 0,
2.76936071771683709295115864629, 3.96033722802915316882524059193, 5.52035129648516313797764764737, 6.13065631909578093325834919161, 7.63939050545091637195351002555, 8.851224139712706448083893835624, 9.429748042845105458622615719434, 10.31767650394093071905287023450, 11.40503845862669404741197697105
