| L(s) = 1 | − 5-s − 11-s − 2·13-s + 6·17-s + 4·19-s + 25-s − 6·29-s + 8·31-s + 6·37-s − 10·41-s + 4·43-s − 8·47-s − 7·49-s + 10·53-s + 55-s − 12·59-s + 6·61-s + 2·65-s + 4·67-s − 14·73-s + 4·83-s − 6·85-s + 6·89-s − 4·95-s − 14·97-s + 2·101-s + 16·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s − 49-s + 1.37·53-s + 0.134·55-s − 1.56·59-s + 0.768·61-s + 0.248·65-s + 0.488·67-s − 1.63·73-s + 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.410·95-s − 1.42·97-s + 0.199·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.706941781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706941781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79421947007216328503850475497, −7.33352655409029437479580252870, −6.50205487545078144112806337954, −5.65629670719373159910303806571, −5.08403251235291987756288455285, −4.33319488739593599837779530762, −3.37110107215072825477040449310, −2.89133388530523048009756846315, −1.70539864855025020791157306525, −0.65882221835164491685357531788,
0.65882221835164491685357531788, 1.70539864855025020791157306525, 2.89133388530523048009756846315, 3.37110107215072825477040449310, 4.33319488739593599837779530762, 5.08403251235291987756288455285, 5.65629670719373159910303806571, 6.50205487545078144112806337954, 7.33352655409029437479580252870, 7.79421947007216328503850475497
