| 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
−18.01837970894455201742489738300, −17.28929903049588589287654530510, −15.11491872280691135397491079404, −12.63544970288734926812134623887, −12.09816551508132502386464926854, −10.59415006618634737377722912041, −9.160991657259376740167531768935, −7.37876536259149094380643403645, −3.84218785750404691704185362770, −1.62599414104385827620483166558,
0.12456454890181482203260916588, 4.79885007841664451270097154684, 6.71487300838998607594956680918, 8.157228965365230069572082694520, 9.578217167163690641342343011357, 11.36234125638855753635771576048, 13.93495958730859788240508100918, 15.34487941882152089174964484724, 16.27264500832589703228054269683, 17.07128310954985086954591974874
