| L(s) = 1 | + (−19.5 + 33.8i)2-s + (−36.0 + 62.4i)3-s + (−509. − 882. i)4-s − 1.00e3·5-s + (−1.41e3 − 2.44e3i)6-s + (190. + 329. i)7-s + 1.98e4·8-s + (7.23e3 + 1.25e4i)9-s + (1.97e4 − 3.41e4i)10-s + (3.11e4 − 5.39e4i)11-s + 7.34e4·12-s + (−8.48e4 + 5.83e4i)13-s − 1.48e4·14-s + (3.63e4 − 6.29e4i)15-s + (−1.26e5 + 2.19e5i)16-s + (−2.08e5 − 3.61e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.864 + 1.49i)2-s + (−0.257 + 0.445i)3-s + (−0.994 − 1.72i)4-s − 0.720·5-s + (−0.444 − 0.770i)6-s + (0.0299 + 0.0518i)7-s + 1.71·8-s + (0.367 + 0.636i)9-s + (0.623 − 1.07i)10-s + (0.641 − 1.11i)11-s + 1.02·12-s + (−0.823 + 0.566i)13-s − 0.103·14-s + (0.185 − 0.321i)15-s + (−0.483 + 0.838i)16-s + (−0.605 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.149606 - 0.0556838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149606 - 0.0556838i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (8.48e4 - 5.83e4i)T \) |
| good | 2 | \( 1 + (19.5 - 33.8i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (36.0 - 62.4i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + 1.00e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + (-190. - 329. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-3.11e4 + 5.39e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 17 | \( 1 + (2.08e5 + 3.61e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (1.18e5 + 2.05e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.86e5 + 1.01e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-2.97e6 + 5.15e6i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 5.96e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (8.42e6 - 1.45e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (6.53e6 - 1.13e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-9.45e6 - 1.63e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + 4.10e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.90e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + (4.79e7 + 8.30e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.00e8 + 1.74e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-2.73e6 + 4.73e6i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-1.64e8 - 2.85e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.75e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.17e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.59e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.38e8 - 4.12e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (3.12e8 + 5.41e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−17.07128310954985086954591974874, −16.27264500832589703228054269683, −15.34487941882152089174964484724, −13.93495958730859788240508100918, −11.36234125638855753635771576048, −9.578217167163690641342343011357, −8.157228965365230069572082694520, −6.71487300838998607594956680918, −4.79885007841664451270097154684, −0.12456454890181482203260916588,
1.62599414104385827620483166558, 3.84218785750404691704185362770, 7.37876536259149094380643403645, 9.160991657259376740167531768935, 10.59415006618634737377722912041, 12.09816551508132502386464926854, 12.63544970288734926812134623887, 15.11491872280691135397491079404, 17.28929903049588589287654530510, 18.01837970894455201742489738300
