| L(s) = 1 | + (1 − i)2-s − 2i·4-s + (2 + i)5-s + (−2 − 2i)8-s + (3 − i)10-s + (−1 − i)13-s − 4·16-s + (−3 + 3i)17-s + (2 − 4i)20-s + (3 + 4i)25-s − 2·26-s + 4i·29-s + (−4 + 4i)32-s + 6i·34-s + (−7 + 7i)37-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + (0.894 + 0.447i)5-s + (−0.707 − 0.707i)8-s + (0.948 − 0.316i)10-s + (−0.277 − 0.277i)13-s − 16-s + (−0.727 + 0.727i)17-s + (0.447 − 0.894i)20-s + (0.600 + 0.800i)25-s − 0.392·26-s + 0.742i·29-s + (−0.707 + 0.707i)32-s + 1.02i·34-s + (−1.15 + 1.15i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52914 - 0.852555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52914 - 0.852555i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (3 - 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (9 + 9i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (11 + 11i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-13 + 13i)T - 97iT^{2} \) |
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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
−12.69794170135484687304413895935, −11.49029736143249133136551533241, −10.57247539133846836722364304916, −9.870320758800327287005653304896, −8.739973580780035675871218399526, −6.92631001022100348507174626909, −5.96147912093056451593267216430, −4.84360594018672742592642180007, −3.29130633826925225407131880272, −1.92433628645762413952817508959,
2.48247368747939277244217051032, 4.29322535774556801723332538839, 5.36059816736116392293212599542, 6.37224151113724282744104274704, 7.46075314863325815153746791658, 8.758274899783008978368742428221, 9.555475699564994929062637634935, 11.02191249766749448320241458224, 12.18370566131111394162278799771, 12.99030959944853555289996114351
