L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s − 2·13-s − 15-s + 2·17-s − 4·19-s + 23-s + 25-s + 27-s + 2·29-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s + 8·43-s − 45-s + 4·47-s + 2·51-s + 2·53-s − 4·55-s − 4·57-s + 12·59-s + 2·61-s + 2·65-s − 8·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.173662289\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.173662289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.87049018956502, −12.20132150966755, −12.00156702357443, −11.43721275311275, −10.98916850177026, −10.39542311799226, −10.02950764929643, −9.432302421064610, −9.004620817304997, −8.719138923362995, −8.121472641188472, −7.587026816851887, −7.252074374051328, −6.767595518759601, −6.041908187467203, −5.892867445116342, −4.915796854596701, −4.497709755951099, −4.043835895889576, −3.606481774551464, −2.964375347135429, −2.386601744702928, −1.899287705910034, −1.007063741673353, −0.6239238119406341,
0.6239238119406341, 1.007063741673353, 1.899287705910034, 2.386601744702928, 2.964375347135429, 3.606481774551464, 4.043835895889576, 4.497709755951099, 4.915796854596701, 5.892867445116342, 6.041908187467203, 6.767595518759601, 7.252074374051328, 7.587026816851887, 8.121472641188472, 8.719138923362995, 9.004620817304997, 9.432302421064610, 10.02950764929643, 10.39542311799226, 10.98916850177026, 11.43721275311275, 12.00156702357443, 12.20132150966755, 12.87049018956502