L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s + (1.5 + 0.866i)6-s + (−1.73 + i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (1.5 + 2.59i)11-s − 1.73·12-s + (−1.73 − i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s + 3i·17-s − 3i·18-s + 19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s + (0.612 + 0.353i)6-s + (−0.654 + 0.377i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.452 + 0.783i)11-s − 0.499·12-s + (−0.480 − 0.277i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s + 0.727i·17-s − 0.707i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.541997 + 0.355789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541997 + 0.355789i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.73 - i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.3 + 6i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (4.33 - 2.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.26822729336336100898691108530, −10.27184914356865221940373154970, −9.427609885370491290416994013722, −8.444409368832258556483158957570, −7.44485240076760961329092586202, −6.73515964007180646489280361305, −5.90697626153175983735823853280, −4.82578862102172239085737852461, −2.87223931124119706554119580590, −1.40218000499110217624621404616,
0.56612770117990957153726409168, 2.87336412358318325695092173203, 3.88572963683494774925813679722, 5.07766923778108887773901744574, 6.34195759900824858139467788264, 7.14835148491112182791744129582, 8.594322741819719972115591621133, 9.278938249927048248330021253212, 10.07070282343605233689656504933, 10.77251430643882653686125306908