| L(s) = 1 | + (−4.57 + 3.32i)2-s + (−16.9 + 23.3i)3-s + (9.88 − 30.4i)4-s + 12.8·5-s − 163. i·6-s + (−163. + 503. i)7-s + (55.9 + 172. i)8-s + (−32.3 − 99.6i)9-s + (−58.7 + 42.6i)10-s + (−1.68e3 − 546. i)11-s + (543. + 747. i)12-s + (−1.57e3 + 2.16e3i)13-s + (−925. − 2.84e3i)14-s + (−217. + 299. i)15-s + (−828. − 601. i)16-s + (3.01e3 − 978. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.628 + 0.865i)3-s + (0.154 − 0.475i)4-s + 0.102·5-s − 0.756i·6-s + (−0.476 + 1.46i)7-s + (0.109 + 0.336i)8-s + (−0.0443 − 0.136i)9-s + (−0.0587 + 0.0426i)10-s + (−1.26 − 0.410i)11-s + (0.314 + 0.432i)12-s + (−0.716 + 0.986i)13-s + (−0.337 − 1.03i)14-s + (−0.0645 + 0.0888i)15-s + (−0.202 − 0.146i)16-s + (0.613 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0337 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0337 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.101421 - 0.104905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101421 - 0.104905i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.57 - 3.32i)T \) |
| 31 | \( 1 + (-7.58e3 - 2.88e4i)T \) |
| good | 3 | \( 1 + (16.9 - 23.3i)T + (-225. - 693. i)T^{2} \) |
| 5 | \( 1 - 12.8T + 1.56e4T^{2} \) |
| 7 | \( 1 + (163. - 503. i)T + (-9.51e4 - 6.91e4i)T^{2} \) |
| 11 | \( 1 + (1.68e3 + 546. i)T + (1.43e6 + 1.04e6i)T^{2} \) |
| 13 | \( 1 + (1.57e3 - 2.16e3i)T + (-1.49e6 - 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-3.01e3 + 978. i)T + (1.95e7 - 1.41e7i)T^{2} \) |
| 19 | \( 1 + (-9.31e3 + 6.76e3i)T + (1.45e7 - 4.47e7i)T^{2} \) |
| 23 | \( 1 + (-1.50e4 + 4.88e3i)T + (1.19e8 - 8.70e7i)T^{2} \) |
| 29 | \( 1 + (2.21e4 + 3.04e4i)T + (-1.83e8 + 5.65e8i)T^{2} \) |
| 37 | \( 1 + 7.37e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (5.89e4 - 4.28e4i)T + (1.46e9 - 4.51e9i)T^{2} \) |
| 43 | \( 1 + (1.63e4 + 2.25e4i)T + (-1.95e9 + 6.01e9i)T^{2} \) |
| 47 | \( 1 + (-1.54e5 - 1.12e5i)T + (3.33e9 + 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-7.93e3 + 2.57e3i)T + (1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (8.29e4 + 6.02e4i)T + (1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 - 7.19e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.41e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (1.24e4 + 3.83e4i)T + (-1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (6.34e5 + 2.06e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-8.81e4 + 2.86e4i)T + (1.96e11 - 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-6.13e5 - 8.43e5i)T + (-1.01e11 + 3.10e11i)T^{2} \) |
| 89 | \( 1 + (7.48e5 + 2.43e5i)T + (4.02e11 + 2.92e11i)T^{2} \) |
| 97 | \( 1 + (2.62e5 - 8.07e5i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
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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
−15.13343411160756025192093135323, −13.58316818986792473632076328355, −12.05137914070430685214525902588, −11.05396081887438755994715363667, −9.784271621699240044875363373895, −9.088663865204117366183199884121, −7.48137341802328480474132066644, −5.76991816758870513166026140786, −5.03774721512623479635924797004, −2.59980050255239597048527010853,
0.084128461368740594793293632087, 1.25301978220246287696486929162, 3.32662791764844200591424817001, 5.48673095662510156525937941555, 7.27189628431144450511718111571, 7.64250286758722709441101483853, 9.846164912912095475219472300469, 10.45984910405805452562378370052, 11.84458476973494356364278470987, 12.87099650387474291408291446775
