| L(s) = 1 | + (19.7 − 60.8i)2-s + (697. − 506. i)3-s + (−3.31e3 − 2.40e3i)4-s + (7.80e3 + 2.40e4i)5-s + (−1.70e4 − 5.24e4i)6-s + (9.62e3 + 6.99e3i)7-s + (−2.12e5 + 1.54e5i)8-s + (−2.62e5 + 8.09e5i)9-s + 1.61e6·10-s + (−5.75e6 + 1.17e6i)11-s − 3.53e6·12-s + (−7.65e6 + 2.35e7i)13-s + (6.16e5 − 4.47e5i)14-s + (1.76e7 + 1.27e7i)15-s + (5.18e6 + 1.59e7i)16-s + (−2.05e7 − 6.32e7i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.552 − 0.401i)3-s + (−0.404 − 0.293i)4-s + (0.223 + 0.687i)5-s + (−0.149 − 0.459i)6-s + (0.0309 + 0.0224i)7-s + (−0.286 + 0.207i)8-s + (−0.164 + 0.507i)9-s + 0.511·10-s + (−0.979 + 0.200i)11-s − 0.341·12-s + (−0.439 + 1.35i)13-s + (0.0218 − 0.0158i)14-s + (0.399 + 0.290i)15-s + (0.0772 + 0.237i)16-s + (−0.206 − 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.427992773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427992773\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.7 + 60.8i)T \) |
| 11 | \( 1 + (5.75e6 - 1.17e6i)T \) |
| good | 3 | \( 1 + (-697. + 506. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-7.80e3 - 2.40e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-9.62e3 - 6.99e3i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (7.65e6 - 2.35e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (2.05e7 + 6.32e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (1.68e8 - 1.22e8i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 + 1.61e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-3.74e9 - 2.72e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (5.65e8 - 1.73e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-7.48e9 - 5.43e9i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (1.29e10 - 9.39e9i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 2.18e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (8.88e10 - 6.45e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-8.34e10 + 2.56e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (3.32e11 + 2.41e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (4.00e10 + 1.23e11i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 + 4.98e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-5.76e11 - 1.77e12i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (4.68e11 + 3.40e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (5.64e11 - 1.73e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-6.96e11 - 2.14e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 + 4.65e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (1.11e12 - 3.43e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−14.60496239680575091694976752040, −13.88441379172988090475134196175, −12.65585084676231514537462290348, −11.17077323049105934304072708321, −9.990050664676222360246268424305, −8.378423140147815340482038539434, −6.80731381029142291201695515650, −4.80913454079897280468886037661, −2.85447486329898169364148001728, −1.90693256466155299953649297056,
0.39739188759047575783305581173, 2.87420747287895641631513204523, 4.57168254444549427152964702207, 5.95357281526374727827129997463, 7.918928329274212338006446736956, 8.933441097047794427995106904284, 10.36447289433978032140142260926, 12.49621641247325859002123170610, 13.45464997734587322013964823901, 14.95141954046791230519864891608
