L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (1.89 + 1.19i)5-s − 1.00i·6-s + (2.48 − 2.48i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (2.18 − 0.496i)10-s + 3.08i·11-s + (−0.707 − 0.707i)12-s + (0.986 − 0.986i)13-s − 3.51i·14-s + (2.18 − 0.496i)15-s − 1.00·16-s + (1.61 + 1.61i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (0.846 + 0.532i)5-s − 0.408i·6-s + (0.938 − 0.938i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.689 − 0.156i)10-s + 0.930i·11-s + (−0.204 − 0.204i)12-s + (0.273 − 0.273i)13-s − 0.938i·14-s + (0.562 − 0.128i)15-s − 0.250·16-s + (0.390 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.44864 - 1.66583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.44864 - 1.66583i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.89 - 1.19i)T \) |
| 31 | \( 1 + (-0.442 + 5.55i)T \) |
good | 7 | \( 1 + (-2.48 + 2.48i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.08iT - 11T^{2} \) |
| 13 | \( 1 + (-0.986 + 0.986i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.61 - 1.61i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.13iT - 19T^{2} \) |
| 23 | \( 1 + (2.98 - 2.98i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 37 | \( 1 + (-0.671 - 0.671i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + (1.90 - 1.90i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.635 - 0.635i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.01 - 4.01i)T - 53iT^{2} \) |
| 59 | \( 1 + 7.25iT - 59T^{2} \) |
| 61 | \( 1 + 1.31iT - 61T^{2} \) |
| 67 | \( 1 + (0.962 - 0.962i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 + (-2.24 + 2.24i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 + (-7.83 + 7.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.895T + 89T^{2} \) |
| 97 | \( 1 + (3.59 - 3.59i)T - 97iT^{2} \) |
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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
−9.978835868717398966123192825581, −9.375090125942831010480611569449, −8.048263764025196643859198664406, −7.37370443277851832414247601503, −6.47005842778823491246434458651, −5.47991975672051504670825022467, −4.45634522450016789861706362302, −3.45928894391149704375229579283, −2.20163078895318925661338905617, −1.40017689010994449721030222189,
1.73044838452346630285920856519, 2.87525855271756503876809153059, 4.11906326819547061203614064290, 5.20571341955081274110719141023, 5.60926658710532851094546127870, 6.59248405244408144858853882507, 7.986080411242470373345439744696, 8.533192766111001574408511142020, 9.122234135944300046911518470748, 10.10062367296600492661947556886