| L(s) = 1 | + (−25.8 + 18.8i)2-s + (−84.0 + 258. i)3-s + (316. − 973. i)4-s + (3.82e3 − 5.84e3i)5-s + (−2.68e3 − 8.27e3i)6-s + 5.43e4·7-s + (1.01e4 + 3.11e4i)8-s + (8.35e4 + 6.06e4i)9-s + (1.08e4 + 2.23e5i)10-s + (1.52e5 − 1.10e5i)11-s + (2.25e5 + 1.63e5i)12-s + (−5.02e4 − 3.64e4i)13-s + (−1.40e6 + 1.02e6i)14-s + (1.18e6 + 1.48e6i)15-s + (−8.48e5 − 6.16e5i)16-s + (−2.39e6 − 7.37e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.199 + 0.614i)3-s + (0.154 − 0.475i)4-s + (0.547 − 0.836i)5-s + (−0.141 − 0.434i)6-s + 1.22·7-s + (0.109 + 0.336i)8-s + (0.471 + 0.342i)9-s + (0.0343 + 0.706i)10-s + (0.284 − 0.207i)11-s + (0.261 + 0.189i)12-s + (−0.0375 − 0.0272i)13-s + (−0.699 + 0.508i)14-s + (0.404 + 0.503i)15-s + (−0.202 − 0.146i)16-s + (−0.409 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.74361 - 0.358448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74361 - 0.358448i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (25.8 - 18.8i)T \) |
| 5 | \( 1 + (-3.82e3 + 5.84e3i)T \) |
| good | 3 | \( 1 + (84.0 - 258. i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 7 | \( 1 - 5.43e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + (-1.52e5 + 1.10e5i)T + (8.81e10 - 2.71e11i)T^{2} \) |
| 13 | \( 1 + (5.02e4 + 3.64e4i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (2.39e6 + 7.37e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (4.31e6 + 1.32e7i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 + (-8.47e6 + 6.16e6i)T + (2.94e14 - 9.06e14i)T^{2} \) |
| 29 | \( 1 + (1.87e6 - 5.77e6i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-4.31e7 - 1.32e8i)T + (-2.05e16 + 1.49e16i)T^{2} \) |
| 37 | \( 1 + (3.29e8 + 2.39e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (-1.82e8 - 1.32e8i)T + (1.70e17 + 5.23e17i)T^{2} \) |
| 43 | \( 1 - 1.79e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-2.59e8 + 7.99e8i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (-1.00e9 + 3.09e9i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (-4.68e9 - 3.40e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-7.46e9 + 5.42e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 + (-3.06e8 - 9.42e8i)T + (-9.88e19 + 7.17e19i)T^{2} \) |
| 71 | \( 1 + (-6.39e9 + 1.96e10i)T + (-1.86e20 - 1.35e20i)T^{2} \) |
| 73 | \( 1 + (2.35e10 - 1.71e10i)T + (9.69e19 - 2.98e20i)T^{2} \) |
| 79 | \( 1 + (-1.65e10 + 5.09e10i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (-6.37e9 - 1.96e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + (3.64e10 - 2.65e10i)T + (8.57e20 - 2.63e21i)T^{2} \) |
| 97 | \( 1 + (-4.27e9 + 1.31e10i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−13.26845604488801627572295128154, −11.61465405274576167067228237615, −10.59449479379812116159035116042, −9.356089437457382682311552362031, −8.483396763279085169594983286930, −7.04398612818727718268941106710, −5.24440242212234322590055115125, −4.61529314296083049130712196944, −2.01106592531998932412354008231, −0.69287457336591124338144375441,
1.32056997860756109947657413257, 2.08185781539915806181237807967, 3.98999811688593259893373826241, 6.02371455143053451117957234371, 7.21242297984260425269606917548, 8.339234329444059312882815415529, 9.887962919455689628703946068827, 10.87532906309878798830532806721, 11.91563745199360360809892702320, 13.04683846432789232376223248354
