L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.70 + 0.707i)5-s + (−0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (1.70 − 0.707i)10-s − 2i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s − 1.00·18-s + (0.707 − 1.70i)20-s + (1.70 + 1.70i)25-s + (−1.41 − 1.41i)26-s + (−0.707 − 0.292i)29-s + (−0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.70 + 0.707i)5-s + (−0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (1.70 − 0.707i)10-s − 2i·13-s − 1.00·16-s + (0.707 + 0.707i)17-s − 1.00·18-s + (0.707 − 1.70i)20-s + (1.70 + 1.70i)25-s + (−1.41 − 1.41i)26-s + (−0.707 − 0.292i)29-s + (−0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.301742830\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.301742830\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
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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
−8.883921378539557812655561042057, −7.926105206715736973886961858960, −6.73474702068710329299753075403, −6.05519717658780310464426052651, −5.65451841112312950090643056823, −5.06227151101061327695043580417, −3.50938111775423619050128459432, −3.07932074125100982915405324204, −2.25033216888217488763691248948, −1.15154846705967676799574009531,
1.81031732133520589114588799588, 2.39861073835203771922515047313, 3.62944143387515088405518473871, 4.83374003956676857481882806514, 5.12505264609351885513588038136, 5.92205775272567661619324465982, 6.53781226825206361041500755016, 7.29721397138687661503451619894, 8.276430417752901527341011349768, 9.081494220046700045230137892619