| L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (3 + 3i)7-s + (0.707 + 0.707i)8-s − 4.24i·11-s + (3 − 3i)13-s − 4.24·14-s − 1.00·16-s + (−4.24 + 4.24i)17-s + 2i·19-s + (3 + 3i)22-s + (4.24 + 4.24i)23-s + 4.24i·26-s + (3.00 − 3.00i)28-s + 8.48·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.13 + 1.13i)7-s + (0.250 + 0.250i)8-s − 1.27i·11-s + (0.832 − 0.832i)13-s − 1.13·14-s − 0.250·16-s + (−1.02 + 1.02i)17-s + 0.458i·19-s + (0.639 + 0.639i)22-s + (0.884 + 0.884i)23-s + 0.832i·26-s + (0.566 − 0.566i)28-s + 1.57·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13459 + 0.511656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13459 + 0.511656i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.24 - 4.24i)T - 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (4.24 + 4.24i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + (6 + 6i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 6i)T - 73iT^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 + (12 + 12i)T + 97iT^{2} \) |
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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
−11.12484859597084846810382096457, −10.41691304458555711885822477616, −9.026357908353123323691007293185, −8.406998596438304340287373599065, −7.977958256632927353916050913093, −6.34643028794037614256392179607, −5.76811156192630985426305303109, −4.70558025091208987450440556086, −3.02845897639445743033854465535, −1.40599523600476505036494093736,
1.16381502625657574254519404517, 2.50712236879917508316777111661, 4.34312292594236896339646801923, 4.66137205900719789171467481818, 6.73497115525536461807279059158, 7.27378098614353589435576664411, 8.384909946225847808031976332846, 9.177398932681114643091932314707, 10.22225304100640072573837852887, 10.97497678768499833050174191972
