| L(s) = 1 | − 4·5-s + 7-s + 2·11-s − 6·13-s + 4·17-s + 4·19-s + 2·23-s + 11·25-s + 2·29-s − 4·35-s + 2·37-s + 4·43-s + 12·47-s + 49-s + 6·53-s − 8·55-s − 8·59-s + 6·61-s + 24·65-s + 8·67-s + 14·71-s − 2·73-s + 2·77-s − 12·79-s − 4·83-s − 16·85-s − 6·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s + 0.603·11-s − 1.66·13-s + 0.970·17-s + 0.917·19-s + 0.417·23-s + 11/5·25-s + 0.371·29-s − 0.676·35-s + 0.328·37-s + 0.609·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 1.04·59-s + 0.768·61-s + 2.97·65-s + 0.977·67-s + 1.66·71-s − 0.234·73-s + 0.227·77-s − 1.35·79-s − 0.439·83-s − 1.73·85-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122902366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122902366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.954334131307010213005667652434, −9.076045831246850385292735724294, −8.100150186340802045251146455502, −7.47655671664491235528608994140, −6.98585917665527551317684669265, −5.44643327053124053530313487020, −4.59055067570292347348864981302, −3.76740235739525618511594493326, −2.73919662589933967809346607483, −0.831916978658255590031556900046,
0.831916978658255590031556900046, 2.73919662589933967809346607483, 3.76740235739525618511594493326, 4.59055067570292347348864981302, 5.44643327053124053530313487020, 6.98585917665527551317684669265, 7.47655671664491235528608994140, 8.100150186340802045251146455502, 9.076045831246850385292735724294, 9.954334131307010213005667652434
