| L(s) = 1 | + (−50.6 − 50.6i)2-s + (910. + 910. i)3-s − 1.12e4i·4-s + 1.27e5i·5-s − 9.22e4i·6-s − 1.54e6·7-s + (−1.40e6 + 1.40e6i)8-s − 3.12e6i·9-s + (6.46e6 − 6.46e6i)10-s + (1.89e6 + 1.89e6i)11-s + (1.02e7 − 1.02e7i)12-s − 7.31e7i·13-s + (7.85e7 + 7.85e7i)14-s + (−1.16e8 + 1.16e8i)15-s − 4.23e7·16-s + (1.23e8 + 1.23e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.395 − 0.395i)2-s + (0.416 + 0.416i)3-s − 0.686i·4-s + 1.63i·5-s − 0.329i·6-s − 1.88·7-s + (−0.667 + 0.667i)8-s − 0.653i·9-s + (0.646 − 0.646i)10-s + (0.0972 + 0.0972i)11-s + (0.285 − 0.285i)12-s − 1.16i·13-s + (0.744 + 0.744i)14-s + (−0.680 + 0.680i)15-s − 0.157·16-s + (0.300 + 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(1.162275543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162275543\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-1.05e10 - 1.36e10i)T \) |
| good | 2 | \( 1 + (50.6 + 50.6i)T + 1.63e4iT^{2} \) |
| 3 | \( 1 + (-910. - 910. i)T + 4.78e6iT^{2} \) |
| 5 | \( 1 - 1.27e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 1.54e6T + 6.78e11T^{2} \) |
| 11 | \( 1 + (-1.89e6 - 1.89e6i)T + 3.79e14iT^{2} \) |
| 13 | \( 1 + 7.31e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-1.23e8 - 1.23e8i)T + 1.68e17iT^{2} \) |
| 19 | \( 1 + (-1.31e7 - 1.31e7i)T + 7.99e17iT^{2} \) |
| 23 | \( 1 - 3.45e9T + 1.15e19T^{2} \) |
| 31 | \( 1 + (-2.10e10 - 2.10e10i)T + 7.56e20iT^{2} \) |
| 37 | \( 1 + (3.02e9 - 3.02e9i)T - 9.01e21iT^{2} \) |
| 41 | \( 1 + (-2.27e11 + 2.27e11i)T - 3.79e22iT^{2} \) |
| 43 | \( 1 + (2.66e10 + 2.66e10i)T + 7.38e22iT^{2} \) |
| 47 | \( 1 + (-3.95e11 + 3.95e11i)T - 2.56e23iT^{2} \) |
| 53 | \( 1 + 4.44e11T + 1.37e24T^{2} \) |
| 59 | \( 1 + 2.41e12T + 6.19e24T^{2} \) |
| 61 | \( 1 + (1.51e12 + 1.51e12i)T + 9.87e24iT^{2} \) |
| 67 | \( 1 + 8.36e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 - 7.18e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + (6.97e12 - 6.97e12i)T - 1.22e26iT^{2} \) |
| 79 | \( 1 + (1.62e13 + 1.62e13i)T + 3.68e26iT^{2} \) |
| 83 | \( 1 - 4.05e13T + 7.36e26T^{2} \) |
| 89 | \( 1 + (-1.71e13 - 1.71e13i)T + 1.95e27iT^{2} \) |
| 97 | \( 1 + (-9.87e13 + 9.87e13i)T - 6.52e27iT^{2} \) |
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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
−14.02507878280844313142581696085, −12.40955335312674850263852747116, −10.62718217873405320839112408002, −10.12666534527968871115464549920, −9.103809981722565912988442805809, −6.89098769220991735254429703819, −6.00057503411228447578721338130, −3.31663416181300866602337938445, −2.82959817962710306078336452474, −0.53149354882263006377393029037,
0.823746460501052053704091742700, 2.77436043598251561645573832425, 4.35751321393225649762058113916, 6.32003243684736072834219052947, 7.64664960051097571936796469606, 8.901277858776228242103539437380, 9.520935563342876477253301060727, 12.02857520491956348624924332929, 12.94881872390195790442346553915, 13.52887089567969885434582614033
