L(s) = 1 | + i·2-s − 4-s + (−1.36 − 0.728i)5-s − i·8-s + (0.831 + 0.555i)9-s + (0.728 − 1.36i)10-s + (1.08 − 0.216i)13-s + 16-s + (1.38 + 1.38i)17-s + (−0.555 + 0.831i)18-s + (1.36 + 0.728i)20-s + (0.772 + 1.15i)25-s + (0.216 + 1.08i)26-s + (−1.17 + 0.785i)29-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (−1.36 − 0.728i)5-s − i·8-s + (0.831 + 0.555i)9-s + (0.728 − 1.36i)10-s + (1.08 − 0.216i)13-s + 16-s + (1.38 + 1.38i)17-s + (−0.555 + 0.831i)18-s + (1.36 + 0.728i)20-s + (0.772 + 1.15i)25-s + (0.216 + 1.08i)26-s + (−1.17 + 0.785i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8175358000\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8175358000\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 257 | \( 1 + T \) |
good | 3 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 5 | \( 1 + (1.36 + 0.728i)T + (0.555 + 0.831i)T^{2} \) |
| 7 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-1.08 + 0.216i)T + (0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 1.38i)T + iT^{2} \) |
| 19 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (1.17 - 0.785i)T + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 37 | \( 1 + (-0.151 + 1.53i)T + (-0.980 - 0.195i)T^{2} \) |
| 41 | \( 1 + (-0.728 + 0.598i)T + (0.195 - 0.980i)T^{2} \) |
| 43 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 47 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 53 | \( 1 + (-0.598 - 1.11i)T + (-0.555 + 0.831i)T^{2} \) |
| 59 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (-1.81 + 0.360i)T + (0.923 - 0.382i)T^{2} \) |
| 67 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 73 | \( 1 + (1.53 + 0.636i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 89 | \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \) |
| 97 | \( 1 + (0.273 - 0.512i)T + (-0.555 - 0.831i)T^{2} \) |
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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
−10.26536840025289190845690932224, −9.078314210260185909283087183510, −8.456352918008466695934031050975, −7.68671465577390539244574260674, −7.30513246819878462869112034306, −5.94445372135810752080415484094, −5.22918036526820309405737398946, −3.94230292690645847036779590057, −3.83600179223693207630622167879, −1.21694136403485106242364710213,
1.06743200876436915685386767151, 2.83317436680373887543639993497, 3.69096319399655838935129076157, 4.22561947372270591140311127065, 5.49937122880090942753858536383, 6.80365421604406798434057565746, 7.62725983227158252728809438057, 8.335368064950826428263128492065, 9.435692159973406863177319692189, 10.04181246497658773908174031871