L(s) = 1 | + (−1.22 − 0.707i)2-s + (−2.09 − 3.62i)3-s + (0.999 + 1.73i)4-s + 5.91i·6-s + (−6.63 + 2.24i)7-s − 2.82i·8-s + (−4.24 + 7.34i)9-s + (6.62 + 11.4i)11-s + (4.18 − 7.24i)12-s + 5.49·13-s + (9.70 + 1.94i)14-s + (−2.00 + 3.46i)16-s + (6.77 + 11.7i)17-s + (10.3 − 6.00i)18-s + (0.621 + 0.358i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.696 − 1.20i)3-s + (0.249 + 0.433i)4-s + 0.985i·6-s + (−0.947 + 0.320i)7-s − 0.353i·8-s + (−0.471 + 0.816i)9-s + (0.601 + 1.04i)11-s + (0.348 − 0.603i)12-s + 0.422·13-s + (0.693 + 0.138i)14-s + (−0.125 + 0.216i)16-s + (0.398 + 0.690i)17-s + (0.577 − 0.333i)18-s + (0.0327 + 0.0188i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.765881 - 0.148951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.765881 - 0.148951i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.63 - 2.24i)T \) |
good | 3 | \( 1 + (2.09 + 3.62i)T + (-4.5 + 7.79i)T^{2} \) |
| 11 | \( 1 + (-6.62 - 11.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 5.49T + 169T^{2} \) |
| 17 | \( 1 + (-6.77 - 11.7i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.96 - 1.13i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 20.4T + 841T^{2} \) |
| 31 | \( 1 + (-21.3 + 12.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-56.2 - 32.4i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.48iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-23.8 + 41.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-19.0 + 11.0i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-72.5 + 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.3 - 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (80.2 - 46.3i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-65.4 - 113. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (38.1 - 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 107.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-145. - 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.5T + 9.40e3T^{2} \) |
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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
−11.50594528558110860073379755395, −10.22288525232290166474716632342, −9.491215564431651194217822144089, −8.355336305080321668578407361733, −7.27052699365332947399336163106, −6.58733264269990239377190685600, −5.73515535453267331960535839129, −3.89592591665607335547653924097, −2.28820112531440924315362471350, −1.01227755775309271203189754409,
0.62805907411968647615959529145, 3.21091147338346473225153187818, 4.32324009006454798552347896956, 5.66594407896339443561376225548, 6.29628346518633059334018131242, 7.52234344733391313413972521135, 8.888143712180021507621185229567, 9.518739509362285191194078731999, 10.31647004299700140680143112532, 11.08109347776999743623508720958