L(s) = 1 | + (−1.39 + 0.204i)2-s + (0.875 + 1.49i)3-s + (1.91 − 0.573i)4-s + (−0.5 + 0.866i)5-s + (−1.53 − 1.91i)6-s + (−1.05 + 0.608i)7-s + (−2.56 + 1.19i)8-s + (−1.46 + 2.61i)9-s + (0.522 − 1.31i)10-s + (−2.23 + 1.29i)11-s + (2.53 + 2.36i)12-s + (−4.16 − 2.40i)13-s + (1.34 − 1.06i)14-s + (−1.73 + 0.0107i)15-s + (3.34 − 2.19i)16-s + 5.97i·17-s + ⋯ |
L(s) = 1 | + (−0.989 + 0.144i)2-s + (0.505 + 0.862i)3-s + (0.958 − 0.286i)4-s + (−0.223 + 0.387i)5-s + (−0.625 − 0.780i)6-s + (−0.398 + 0.229i)7-s + (−0.906 + 0.422i)8-s + (−0.489 + 0.872i)9-s + (0.165 − 0.415i)10-s + (−0.673 + 0.389i)11-s + (0.731 + 0.681i)12-s + (−1.15 − 0.667i)13-s + (0.360 − 0.285i)14-s + (−0.447 + 0.00278i)15-s + (0.835 − 0.549i)16-s + 1.44i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134471 + 0.624768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134471 + 0.624768i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.204i)T \) |
| 3 | \( 1 + (-0.875 - 1.49i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (1.05 - 0.608i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.23 - 1.29i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.16 + 2.40i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.97iT - 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + (0.866 - 1.50i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.85 + 6.67i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.30 - 2.48i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.33iT - 37T^{2} \) |
| 41 | \( 1 + (2.00 + 1.16i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.97 + 3.41i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.0729T + 53T^{2} \) |
| 59 | \( 1 + (-5.32 - 3.07i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (9.89 - 5.71i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 4.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + (-9.17 + 5.29i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.02 + 2.90i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11.5iT - 89T^{2} \) |
| 97 | \( 1 + (-4.84 - 8.39i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.55418926710070867848039447312, −10.41113806761461119748494989546, −10.06350236371842235731586708620, −9.235212376597598927717455794088, −8.032598959853730315917297062874, −7.60397958695661695905296439472, −6.16127874724875812854447197505, −5.03993641008441958612710611056, −3.37881379136413953107252471300, −2.36627203125984951531078046471,
0.52742383035483205803202953160, 2.26204732772845716877629573982, 3.34090704999365734859531437809, 5.29600709730843200228082991230, 6.78954084445086358692690038653, 7.37842506179264222589416779923, 8.181126153047535714907509947703, 9.279045892205488667144179805167, 9.725283611140203736667009222688, 11.12996765689410453653136096951