L(s) = 1 | + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.956 − 0.290i)7-s + (−0.290 + 0.956i)8-s + (0.471 + 0.881i)9-s + (0.0584 − 0.0788i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−0.0727 − 0.0659i)22-s + (0.172 + 1.75i)23-s + (0.773 − 0.634i)25-s + (−0.995 + 0.0980i)28-s + (0.0988 − 0.666i)29-s + (0.471 − 0.881i)32-s + ⋯ |
L(s) = 1 | + (0.0980 − 0.995i)2-s + (−0.980 − 0.195i)4-s + (0.956 − 0.290i)7-s + (−0.290 + 0.956i)8-s + (0.471 + 0.881i)9-s + (0.0584 − 0.0788i)11-s + (−0.195 − 0.980i)14-s + (0.923 + 0.382i)16-s + (0.923 − 0.382i)18-s + (−0.0727 − 0.0659i)22-s + (0.172 + 1.75i)23-s + (0.773 − 0.634i)25-s + (−0.995 + 0.0980i)28-s + (0.0988 − 0.666i)29-s + (0.471 − 0.881i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.243912931\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.243912931\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0980 + 0.995i)T \) |
| 7 | \( 1 + (-0.956 + 0.290i)T \) |
good | 3 | \( 1 + (-0.471 - 0.881i)T^{2} \) |
| 5 | \( 1 + (-0.773 + 0.634i)T^{2} \) |
| 11 | \( 1 + (-0.0584 + 0.0788i)T + (-0.290 - 0.956i)T^{2} \) |
| 13 | \( 1 + (-0.634 + 0.773i)T^{2} \) |
| 17 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 19 | \( 1 + (0.995 - 0.0980i)T^{2} \) |
| 23 | \( 1 + (-0.172 - 1.75i)T + (-0.980 + 0.195i)T^{2} \) |
| 29 | \( 1 + (-0.0988 + 0.666i)T + (-0.956 - 0.290i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.800 - 0.882i)T + (-0.0980 + 0.995i)T^{2} \) |
| 41 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (1.01 + 1.69i)T + (-0.471 + 0.881i)T^{2} \) |
| 47 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 53 | \( 1 + (-0.0713 - 0.480i)T + (-0.956 + 0.290i)T^{2} \) |
| 59 | \( 1 + (-0.634 - 0.773i)T^{2} \) |
| 61 | \( 1 + (-0.881 + 0.471i)T^{2} \) |
| 67 | \( 1 + (0.249 + 0.997i)T + (-0.881 + 0.471i)T^{2} \) |
| 71 | \( 1 + (0.979 + 0.523i)T + (0.555 + 0.831i)T^{2} \) |
| 73 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 79 | \( 1 + (0.704 + 1.05i)T + (-0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (0.0980 + 0.995i)T^{2} \) |
| 89 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 97 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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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
−9.517881866770412508925434617716, −8.590604111975253151373600740470, −7.938064169122681135282191321873, −7.19576616559327135444428353976, −5.82299241659177563217834681506, −4.95847404927474110727555364298, −4.41070798455180576852066870111, −3.37227595074854146029098504693, −2.18814868588096574437376678880, −1.31902674300066055393665556366,
1.21033889256746876829880759413, 2.89629648767540922176861756842, 4.12711976368928024758376039959, 4.73981532603703571479717447829, 5.61929667954453477423742783729, 6.54304531798747134074531553248, 7.10576380716618939457273306121, 8.046991837868724973825572022007, 8.692572046715962009963757328925, 9.312876360492640052682581008289