| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 2·11-s + 6·13-s − 3·15-s − 2·17-s + 9·23-s + 25-s − 9·27-s + 3·29-s + 2·31-s − 6·33-s + 8·37-s − 18·39-s − 5·41-s − 43-s + 6·45-s + 8·47-s + 6·51-s + 4·53-s + 2·55-s − 8·59-s − 7·61-s + 6·65-s + 3·67-s − 27·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.774·15-s − 0.485·17-s + 1.87·23-s + 1/5·25-s − 1.73·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s + 1.31·37-s − 2.88·39-s − 0.780·41-s − 0.152·43-s + 0.894·45-s + 1.16·47-s + 0.840·51-s + 0.549·53-s + 0.269·55-s − 1.04·59-s − 0.896·61-s + 0.744·65-s + 0.366·67-s − 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.388511446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388511446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.694241677182228172269756154818, −7.44163518009335683346060783404, −6.68764982840299092997554232374, −6.19980485768653950501230229858, −5.67902275006040205840033693057, −4.78319451656118645556724774548, −4.18265651043873632086354730598, −3.02914700582274231510062881672, −1.49229063377086535263762304039, −0.837009286959784474979743136837,
0.837009286959784474979743136837, 1.49229063377086535263762304039, 3.02914700582274231510062881672, 4.18265651043873632086354730598, 4.78319451656118645556724774548, 5.67902275006040205840033693057, 6.19980485768653950501230229858, 6.68764982840299092997554232374, 7.44163518009335683346060783404, 8.694241677182228172269756154818
