| L(s) = 1 | + (8.09 − 5.20i)2-s + (2.21 − 15.4i)3-s + (11.8 − 26.0i)4-s + (36.3 + 123. i)5-s + (−62.3 − 136. i)6-s + (399. − 346. i)7-s + (48.4 + 336. i)8-s + (−233. − 68.4i)9-s + (938. + 813. i)10-s + (1.34e3 − 2.09e3i)11-s + (−375. − 241. i)12-s + (−849. + 980. i)13-s + (1.43e3 − 4.87e3i)14-s + (1.99e3 − 286. i)15-s + (3.34e3 + 3.86e3i)16-s + (6.35e3 − 2.90e3i)17-s + ⋯ |
| L(s) = 1 | + (1.01 − 0.650i)2-s + (0.0821 − 0.571i)3-s + (0.185 − 0.407i)4-s + (0.290 + 0.990i)5-s + (−0.288 − 0.631i)6-s + (1.16 − 1.00i)7-s + (0.0946 + 0.657i)8-s + (−0.319 − 0.0939i)9-s + (0.938 + 0.813i)10-s + (1.00 − 1.57i)11-s + (−0.217 − 0.139i)12-s + (−0.386 + 0.446i)13-s + (0.522 − 1.77i)14-s + (0.589 − 0.0848i)15-s + (0.816 + 0.942i)16-s + (1.29 − 0.590i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.15749 - 2.05688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15749 - 2.05688i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.21 + 15.4i)T \) |
| 23 | \( 1 + (1.19e4 - 2.50e3i)T \) |
| good | 2 | \( 1 + (-8.09 + 5.20i)T + (26.5 - 58.2i)T^{2} \) |
| 5 | \( 1 + (-36.3 - 123. i)T + (-1.31e4 + 8.44e3i)T^{2} \) |
| 7 | \( 1 + (-399. + 346. i)T + (1.67e4 - 1.16e5i)T^{2} \) |
| 11 | \( 1 + (-1.34e3 + 2.09e3i)T + (-7.35e5 - 1.61e6i)T^{2} \) |
| 13 | \( 1 + (849. - 980. i)T + (-6.86e5 - 4.77e6i)T^{2} \) |
| 17 | \( 1 + (-6.35e3 + 2.90e3i)T + (1.58e7 - 1.82e7i)T^{2} \) |
| 19 | \( 1 + (4.99e3 + 2.28e3i)T + (3.08e7 + 3.55e7i)T^{2} \) |
| 29 | \( 1 + (1.97e3 + 4.32e3i)T + (-3.89e8 + 4.49e8i)T^{2} \) |
| 31 | \( 1 + (-4.01e3 - 2.79e4i)T + (-8.51e8 + 2.50e8i)T^{2} \) |
| 37 | \( 1 + (-2.10e3 + 7.15e3i)T + (-2.15e9 - 1.38e9i)T^{2} \) |
| 41 | \( 1 + (1.01e5 - 2.97e4i)T + (3.99e9 - 2.56e9i)T^{2} \) |
| 43 | \( 1 + (-8.22e4 - 1.18e4i)T + (6.06e9 + 1.78e9i)T^{2} \) |
| 47 | \( 1 - 1.67e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-1.88e3 + 1.63e3i)T + (3.15e9 - 2.19e10i)T^{2} \) |
| 59 | \( 1 + (1.63e5 - 1.88e5i)T + (-6.00e9 - 4.17e10i)T^{2} \) |
| 61 | \( 1 + (2.36e5 - 3.40e4i)T + (4.94e10 - 1.45e10i)T^{2} \) |
| 67 | \( 1 + (-1.64e5 - 2.56e5i)T + (-3.75e10 + 8.22e10i)T^{2} \) |
| 71 | \( 1 + (4.87e5 - 3.13e5i)T + (5.32e10 - 1.16e11i)T^{2} \) |
| 73 | \( 1 + (1.53e5 - 3.36e5i)T + (-9.91e10 - 1.14e11i)T^{2} \) |
| 79 | \( 1 + (3.37e5 + 2.92e5i)T + (3.45e10 + 2.40e11i)T^{2} \) |
| 83 | \( 1 + (4.26e4 - 1.45e5i)T + (-2.75e11 - 1.76e11i)T^{2} \) |
| 89 | \( 1 + (5.40e5 + 7.77e4i)T + (4.76e11 + 1.40e11i)T^{2} \) |
| 97 | \( 1 + (1.13e4 + 3.87e4i)T + (-7.00e11 + 4.50e11i)T^{2} \) |
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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
−13.71506725080978758847914072646, −12.07223736502066620774971361874, −11.34513134073754766463424964925, −10.49320517068831642439793687251, −8.468052410322723006642356323122, −7.22280602436672545528086595138, −5.82307962706618829604301884294, −4.16886113933262940351123322052, −2.92548987547063330703866734218, −1.34566003842667974298148786765,
1.70590072856769960367163723652, 4.17809164550118701754089359454, 5.01565350641605906649809706869, 5.92900738601405694930879521389, 7.78632997776625863407994453945, 9.093733370451544766502096305618, 10.14018829822798488956749359953, 12.17091009291751374908968332756, 12.45123219635644384214488221555, 14.08985396058648469425983881824
