L(s) = 1 | + (0.631 − 2.35i)2-s + (−0.775 − 1.54i)3-s + (−3.42 − 1.97i)4-s + (1.90 + 1.16i)5-s + (−4.13 + 0.849i)6-s + (1.82 + 1.91i)7-s + (−3.36 + 3.36i)8-s + (−1.79 + 2.40i)9-s + (3.94 − 3.76i)10-s + (−3.08 − 1.77i)11-s + (−0.406 + 6.83i)12-s + (1.28 + 1.28i)13-s + (5.67 − 3.08i)14-s + (0.323 − 3.85i)15-s + (1.85 + 3.21i)16-s + (2.95 − 0.792i)17-s + ⋯ |
L(s) = 1 | + (0.446 − 1.66i)2-s + (−0.447 − 0.894i)3-s + (−1.71 − 0.987i)4-s + (0.853 + 0.520i)5-s + (−1.68 + 0.346i)6-s + (0.688 + 0.725i)7-s + (−1.19 + 1.19i)8-s + (−0.599 + 0.800i)9-s + (1.24 − 1.18i)10-s + (−0.928 − 0.536i)11-s + (−0.117 + 1.97i)12-s + (0.356 + 0.356i)13-s + (1.51 − 0.823i)14-s + (0.0834 − 0.996i)15-s + (0.463 + 0.803i)16-s + (0.717 − 0.192i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332454 - 1.10914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332454 - 1.10914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.775 + 1.54i)T \) |
| 5 | \( 1 + (-1.90 - 1.16i)T \) |
| 7 | \( 1 + (-1.82 - 1.91i)T \) |
good | 2 | \( 1 + (-0.631 + 2.35i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (3.08 + 1.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.28 - 1.28i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.95 + 0.792i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.331 - 0.191i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.45 + 0.658i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.00 - 1.34i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (0.335 + 0.335i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.751 + 2.80i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.815 + 3.04i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.81 - 6.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.31 + 12.3i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (-3.17 + 0.849i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.21 + 1.85i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 + 0.973i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.51 + 2.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 - 10.3i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.19785307736018093361497087389, −12.21157362013961927653077685035, −11.30156061547881173261835455257, −10.67309338426902417011428620601, −9.427922968134847311291705895132, −7.966014379016901313751153690391, −6.04743256234968011084909853256, −5.09812177866896158162733944307, −2.87373382632760355260274470819, −1.74408700952844118642915398671,
4.16591415841522533582424218687, 5.21517852474534870273623191639, 5.89929424482653592114932325099, 7.43504493554958861740396633594, 8.520037126418119709005001331081, 9.762456489500114655769521971196, 10.78591150360297311406672427294, 12.53520313006248356475749811983, 13.56282598606011821074811397226, 14.41166134932556154608137536394