L(s) = 1 | + (−1.58 − 1.58i)5-s + (−1.23 + 1.23i)7-s + 1.74i·11-s + (0.236 + 0.236i)13-s + (−4.57 − 4.57i)17-s + 6.47i·19-s + (−2.82 + 2.82i)23-s + 5.00i·25-s − 0.333·29-s − 10.4·31-s + 3.90·35-s + (−2.23 + 2.23i)37-s + 7.07i·41-s + (6.47 + 6.47i)43-s + (−4.57 − 4.57i)47-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)5-s + (−0.467 + 0.467i)7-s + 0.527i·11-s + (0.0654 + 0.0654i)13-s + (−1.10 − 1.10i)17-s + 1.48i·19-s + (−0.589 + 0.589i)23-s + 1.00i·25-s − 0.0619·29-s − 1.88·31-s + 0.660·35-s + (−0.367 + 0.367i)37-s + 1.10i·41-s + (0.986 + 0.986i)43-s + (−0.667 − 0.667i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.130243 + 0.344196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.130243 + 0.344196i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.58 + 1.58i)T \) |
good | 7 | \( 1 + (1.23 - 1.23i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.74iT - 11T^{2} \) |
| 13 | \( 1 + (-0.236 - 0.236i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.57 + 4.57i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.47iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.333T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + (2.23 - 2.23i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-6.47 - 6.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.57 + 4.57i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 - 1.52T + 61T^{2} \) |
| 67 | \( 1 + (-10.4 + 10.4i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (9.47 + 9.47i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.52iT - 79T^{2} \) |
| 83 | \( 1 + (7.40 - 7.40i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−10.85049324679830096621969334772, −9.573034590217746157603982702885, −9.203347580350203324888872878281, −8.104930486710204992374771801246, −7.41232453176049669045122213160, −6.30838193319940082923316913220, −5.27514886549352777229831645901, −4.33362756629989360763669491424, −3.31052206200460038856614594526, −1.79198744285935801235893332622,
0.18196381259423937430583536183, 2.34582846566591238371476955803, 3.58121142463470282852155554037, 4.27808234472453182659989376556, 5.72364986337816413305378210150, 6.75479124037697603719502571929, 7.26476103409771531068474486195, 8.421258189829115004863037587842, 9.092634895274976646648332553880, 10.36666565621639351340447626184