| L(s) = 1 | + 3-s − 7-s − 2·9-s − 2·13-s − 3·17-s − 19-s − 21-s − 6·23-s − 5·27-s − 9·29-s + 5·31-s + 5·37-s − 2·39-s + 6·41-s − 8·43-s − 6·47-s − 6·49-s − 3·51-s + 9·53-s − 57-s − 6·59-s + 5·61-s + 2·63-s + 8·67-s − 6·69-s − 9·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 0.962·27-s − 1.67·29-s + 0.898·31-s + 0.821·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.875·47-s − 6/7·49-s − 0.420·51-s + 1.23·53-s − 0.132·57-s − 0.781·59-s + 0.640·61-s + 0.251·63-s + 0.977·67-s − 0.722·69-s − 1.06·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.45983994253487, −13.11430686640055, −12.83294871050622, −12.09400372267054, −11.62837899125581, −11.33145120770775, −10.80136317650638, −10.05063268526703, −9.778776940528628, −9.380440627302408, −8.708674536147519, −8.416643763899920, −7.902610921585267, −7.387276541964794, −6.905934269077471, −6.202691871384420, −5.908433035350357, −5.345322055215448, −4.604897004696870, −4.173200108519507, −3.571616559034079, −3.033372822817481, −2.415552753966000, −2.064434359359539, −1.268534600306928, 0, 0,
1.268534600306928, 2.064434359359539, 2.415552753966000, 3.033372822817481, 3.571616559034079, 4.173200108519507, 4.604897004696870, 5.345322055215448, 5.908433035350357, 6.202691871384420, 6.905934269077471, 7.387276541964794, 7.902610921585267, 8.416643763899920, 8.708674536147519, 9.380440627302408, 9.778776940528628, 10.05063268526703, 10.80136317650638, 11.33145120770775, 11.62837899125581, 12.09400372267054, 12.83294871050622, 13.11430686640055, 13.45983994253487
