L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.144 − 1.72i)3-s − 1.00i·4-s + (1.25 + 1.85i)5-s + (1.32 + 1.11i)6-s + (−2.31 − 2.31i)7-s + (0.707 + 0.707i)8-s + (−2.95 + 0.499i)9-s + (−2.19 − 0.423i)10-s − 0.0826i·11-s + (−1.72 + 0.144i)12-s + (−2.92 + 2.92i)13-s + 3.27·14-s + (3.01 − 2.43i)15-s − 1.00·16-s + (−5.30 + 5.30i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.0834 − 0.996i)3-s − 0.500i·4-s + (0.560 + 0.828i)5-s + (0.539 + 0.456i)6-s + (−0.875 − 0.875i)7-s + (0.250 + 0.250i)8-s + (−0.986 + 0.166i)9-s + (−0.694 − 0.134i)10-s − 0.0249i·11-s + (−0.498 + 0.0417i)12-s + (−0.810 + 0.810i)13-s + 0.875·14-s + (0.778 − 0.627i)15-s − 0.250·16-s + (−1.28 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0179955 + 0.105159i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0179955 + 0.105159i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.144 + 1.72i)T \) |
| 5 | \( 1 + (-1.25 - 1.85i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (2.31 + 2.31i)T + 7iT^{2} \) |
| 11 | \( 1 + 0.0826iT - 11T^{2} \) |
| 13 | \( 1 + (2.92 - 2.92i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.30 - 5.30i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.47iT - 19T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 + (-4.80 - 4.80i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.882iT - 41T^{2} \) |
| 43 | \( 1 + (1.69 - 1.69i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.63 - 7.63i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.26 + 7.26i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 0.876T + 61T^{2} \) |
| 67 | \( 1 + (-0.768 - 0.768i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.70iT - 71T^{2} \) |
| 73 | \( 1 + (0.265 - 0.265i)T - 73iT^{2} \) |
| 79 | \( 1 - 16.8iT - 79T^{2} \) |
| 83 | \( 1 + (8.84 + 8.84i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + (7.30 + 7.30i)T + 97iT^{2} \) |
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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
−10.88152131600235096240245583070, −9.863268007359249106656884522353, −9.161637706293323892201175450720, −8.081729730619936859856898924130, −6.94571534901659666783260959353, −6.78596432510268813076303089780, −6.04374529493639084424667229547, −4.56124592217815603949696571996, −2.95038298095775691675477668208, −1.81101283844967153637370691678,
0.06130160281348716279869745246, 2.25868627599946331486060094087, 3.23693571704913633956808649914, 4.55584531568654148171572815776, 5.43469501358959507088328602010, 6.28618750546522510908077227613, 7.80404405072727829682448800556, 8.814600931421432102455890142469, 9.390178693291344358042828174450, 9.856719605375482412785139979654