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
−10.36666565621639351340447626184, −9.092634895274976646648332553880, −8.421258189829115004863037587842, −7.26476103409771531068474486195, −6.75479124037697603719502571929, −5.72364986337816413305378210150, −4.27808234472453182659989376556, −3.58121142463470282852155554037, −2.34582846566591238371476955803, −0.18196381259423937430583536183,
1.79198744285935801235893332622, 3.31052206200460038856614594526, 4.33362756629989360763669491424, 5.27514886549352777229831645901, 6.30838193319940082923316913220, 7.41232453176049669045122213160, 8.104930486710204992374771801246, 9.203347580350203324888872878281, 9.573034590217746157603982702885, 10.85049324679830096621969334772