L(s) = 1 | + (8.14 + 4.70i)2-s + (28.2 + 48.8i)4-s + (−33.1 + 57.4i)5-s + (−116. + 57.3i)7-s + 229. i·8-s + (−540. + 312. i)10-s + (383. − 221. i)11-s + 437. i·13-s + (−1.21e3 − 79.6i)14-s + (−176. + 306. i)16-s + (476. + 825. i)17-s + (1.93e3 + 1.11e3i)19-s − 3.74e3·20-s + 4.16e3·22-s + (−2.09e3 − 1.21e3i)23-s + ⋯ |
L(s) = 1 | + (1.43 + 0.831i)2-s + (0.881 + 1.52i)4-s + (−0.593 + 1.02i)5-s + (−0.896 + 0.442i)7-s + 1.26i·8-s + (−1.70 + 0.986i)10-s + (0.955 − 0.551i)11-s + 0.717i·13-s + (−1.65 − 0.108i)14-s + (−0.172 + 0.299i)16-s + (0.399 + 0.692i)17-s + (1.22 + 0.709i)19-s − 2.09·20-s + 1.83·22-s + (−0.826 − 0.477i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.22192 + 2.85955i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22192 + 2.85955i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (116. - 57.3i)T \) |
good | 2 | \( 1 + (-8.14 - 4.70i)T + (16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (33.1 - 57.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-383. + 221. i)T + (8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 437. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-476. - 825. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.93e3 - 1.11e3i)T + (1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.09e3 + 1.21e3i)T + (3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 6.38e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (-5.91e3 + 3.41e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (7.60e3 - 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.37e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (9.07e3 - 1.57e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.07e4 + 1.20e4i)T + (2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (2.12e4 + 3.67e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.77e4 + 1.02e4i)T + (4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (290. + 503. i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.26e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (4.74e4 - 2.73e4i)T + (1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.85e4 - 3.22e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 4.46e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.20e4 - 2.09e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 2.89e4iT - 8.58e9T^{2} \) |
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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
−14.34684115496398343112107648565, −13.66980835815161694692009136944, −12.21539623773250183701026667740, −11.60456363293451333420516369600, −9.816155918618539084964725981670, −7.941214543097222959430887925548, −6.62597322550813639372545648420, −6.00710328173230596274242156586, −4.04376331545129285834274477290, −3.11032366563230559026745902432,
1.00258838803318720291137803322, 3.19830972772578849022350720396, 4.32701058317131235031058852999, 5.51433545526382509883247749057, 7.20398980793882200901059834299, 9.136508233598982929982278584942, 10.42058313446164616852283630256, 11.94035393277428760868535588964, 12.31143527663370812008126178215, 13.38543230278101763444257059714