L(s) = 1 | + (29.1 − 29.1i)2-s + (−96.9 + 96.9i)3-s − 675. i·4-s − 1.08e3i·5-s + 5.65e3i·6-s − 1.76e4·7-s + (1.01e4 + 1.01e4i)8-s + 4.02e4i·9-s + (−3.17e4 − 3.17e4i)10-s + (6.61e4 − 6.61e4i)11-s + (6.55e4 + 6.55e4i)12-s + 4.40e5i·13-s + (−5.15e5 + 5.15e5i)14-s + (1.05e5 + 1.05e5i)15-s + 1.28e6·16-s + (−1.42e6 + 1.42e6i)17-s + ⋯ |
L(s) = 1 | + (0.910 − 0.910i)2-s + (−0.399 + 0.399i)3-s − 0.659i·4-s − 0.348i·5-s + 0.727i·6-s − 1.05·7-s + (0.309 + 0.309i)8-s + 0.681i·9-s + (−0.317 − 0.317i)10-s + (0.410 − 0.410i)11-s + (0.263 + 0.263i)12-s + 1.18i·13-s + (−0.957 + 0.957i)14-s + (0.139 + 0.139i)15-s + 1.22·16-s + (−1.00 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.42998 + 0.902746i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42998 + 0.902746i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-5.16e6 + 1.98e7i)T \) |
good | 2 | \( 1 + (-29.1 + 29.1i)T - 1.02e3iT^{2} \) |
| 3 | \( 1 + (96.9 - 96.9i)T - 5.90e4iT^{2} \) |
| 5 | \( 1 + 1.08e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 1.76e4T + 2.82e8T^{2} \) |
| 11 | \( 1 + (-6.61e4 + 6.61e4i)T - 2.59e10iT^{2} \) |
| 13 | \( 1 - 4.40e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (1.42e6 - 1.42e6i)T - 2.01e12iT^{2} \) |
| 19 | \( 1 + (3.09e6 - 3.09e6i)T - 6.13e12iT^{2} \) |
| 23 | \( 1 - 1.03e7T + 4.14e13T^{2} \) |
| 31 | \( 1 + (2.89e7 - 2.89e7i)T - 8.19e14iT^{2} \) |
| 37 | \( 1 + (2.36e7 + 2.36e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + (4.75e6 + 4.75e6i)T + 1.34e16iT^{2} \) |
| 43 | \( 1 + (1.29e8 - 1.29e8i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (2.61e8 + 2.61e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + 4.69e7T + 1.74e17T^{2} \) |
| 59 | \( 1 - 6.63e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (7.34e8 - 7.34e8i)T - 7.13e17iT^{2} \) |
| 67 | \( 1 - 9.94e7iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.81e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + (-1.96e9 - 1.96e9i)T + 4.29e18iT^{2} \) |
| 79 | \( 1 + (-6.17e8 + 6.17e8i)T - 9.46e18iT^{2} \) |
| 83 | \( 1 - 4.18e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + (-7.05e9 + 7.05e9i)T - 3.11e19iT^{2} \) |
| 97 | \( 1 + (-1.10e9 - 1.10e9i)T + 7.37e19iT^{2} \) |
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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
−14.73788164732509520834975622361, −13.36931808062354894549849650841, −12.63559701866984280443237670192, −11.31321124962625774904800487942, −10.37809936025655012161790564170, −8.709820215691365540979789975445, −6.40964199699917048483638345106, −4.77761692021491818467718491670, −3.64745290549731345820339857969, −1.88705027966621263302289128218,
0.48089047363145323493877288896, 3.20383161708784939084395047343, 4.97931922651279942777713147364, 6.57220790582094503345677121127, 6.93617623785304749748006601471, 9.248511904648557925252842244162, 10.91205291236515353414429856436, 12.72343043551194829340535683999, 13.17578866672107319331639224564, 14.86632085329185925459336428148