L(s) = 1 | + (0.956 + 0.290i)2-s + (0.831 + 0.555i)4-s + (−0.773 − 0.634i)7-s + (0.634 + 0.773i)8-s + (0.0980 − 0.995i)9-s + (1.07 − 0.509i)11-s + (−0.555 − 0.831i)14-s + (0.382 + 0.923i)16-s + (0.382 − 0.923i)18-s + (1.17 − 0.174i)22-s + (1.90 − 0.577i)23-s + (−0.881 + 0.471i)25-s + (−0.290 − 0.956i)28-s + (−1.82 + 0.653i)29-s + (0.0980 + 0.995i)32-s + ⋯ |
L(s) = 1 | + (0.956 + 0.290i)2-s + (0.831 + 0.555i)4-s + (−0.773 − 0.634i)7-s + (0.634 + 0.773i)8-s + (0.0980 − 0.995i)9-s + (1.07 − 0.509i)11-s + (−0.555 − 0.831i)14-s + (0.382 + 0.923i)16-s + (0.382 − 0.923i)18-s + (1.17 − 0.174i)22-s + (1.90 − 0.577i)23-s + (−0.881 + 0.471i)25-s + (−0.290 − 0.956i)28-s + (−1.82 + 0.653i)29-s + (0.0980 + 0.995i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.030398685\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.030398685\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.956 - 0.290i)T \) |
| 7 | \( 1 + (0.773 + 0.634i)T \) |
good | 3 | \( 1 + (-0.0980 + 0.995i)T^{2} \) |
| 5 | \( 1 + (0.881 - 0.471i)T^{2} \) |
| 11 | \( 1 + (-1.07 + 0.509i)T + (0.634 - 0.773i)T^{2} \) |
| 13 | \( 1 + (-0.471 + 0.881i)T^{2} \) |
| 17 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 19 | \( 1 + (0.290 + 0.956i)T^{2} \) |
| 23 | \( 1 + (-1.90 + 0.577i)T + (0.831 - 0.555i)T^{2} \) |
| 29 | \( 1 + (1.82 - 0.653i)T + (0.773 - 0.634i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.265 - 1.78i)T + (-0.956 - 0.290i)T^{2} \) |
| 41 | \( 1 + (-0.555 - 0.831i)T^{2} \) |
| 43 | \( 1 + (-0.452 + 0.499i)T + (-0.0980 - 0.995i)T^{2} \) |
| 47 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 53 | \( 1 + (0.0923 + 0.0330i)T + (0.773 + 0.634i)T^{2} \) |
| 59 | \( 1 + (-0.471 - 0.881i)T^{2} \) |
| 61 | \( 1 + (-0.995 - 0.0980i)T^{2} \) |
| 67 | \( 1 + (-0.0659 + 1.34i)T + (-0.995 - 0.0980i)T^{2} \) |
| 71 | \( 1 + (1.95 - 0.192i)T + (0.980 - 0.195i)T^{2} \) |
| 73 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 79 | \( 1 + (0.924 - 0.183i)T + (0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (0.956 - 0.290i)T^{2} \) |
| 89 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 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
−9.317929649443466612809045485220, −8.815729263537130188629326889107, −7.54060680324377187256624392381, −6.86080023842456609235385377897, −6.37261984646165457223771412679, −5.52563045183091122082311434505, −4.39622561375101540860269177971, −3.58030784443890810909283118043, −3.10030257964995933707039328605, −1.36852538111474267944549193473,
1.69520971961813026493590070480, 2.59133477521843441709169141117, 3.64967220949187080329605482082, 4.43591643531319919248561969373, 5.47565230899576142080643900383, 5.98573327322812623840734781135, 7.07087404647991792999342378880, 7.50147457700071518319965363871, 8.926463706702192034631386801332, 9.517238689586608653952014630604