L(s) = 1 | + (−3.24 + 2.34i)2-s + 7.00i·3-s + (5.04 − 15.1i)4-s + 9.28·5-s + (−16.3 − 22.7i)6-s + 54.2i·7-s + (19.1 + 61.0i)8-s + 31.9·9-s + (−30.1 + 21.7i)10-s + 65.2i·11-s + (106. + 35.3i)12-s − 57.2·13-s + (−126. − 175. i)14-s + 65.0i·15-s + (−205. − 153. i)16-s − 431.·17-s + ⋯ |
L(s) = 1 | + (−0.810 + 0.585i)2-s + 0.778i·3-s + (0.315 − 0.949i)4-s + 0.371·5-s + (−0.455 − 0.631i)6-s + 1.10i·7-s + (0.299 + 0.953i)8-s + 0.394·9-s + (−0.301 + 0.217i)10-s + 0.539i·11-s + (0.738 + 0.245i)12-s − 0.338·13-s + (−0.647 − 0.897i)14-s + 0.289i·15-s + (−0.801 − 0.598i)16-s − 1.49·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.149723 + 0.926035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.149723 + 0.926035i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.24 - 2.34i)T \) |
| 19 | \( 1 + 82.8iT \) |
good | 3 | \( 1 - 7.00iT - 81T^{2} \) |
| 5 | \( 1 - 9.28T + 625T^{2} \) |
| 7 | \( 1 - 54.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 65.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 57.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + 431.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 458. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 231.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 195. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.29e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.13e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.22e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.36e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 5.21e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.37e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.23e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.62e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 175. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.13e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.20e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.29e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.47e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.06e4T + 8.85e7T^{2} \) |
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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
−14.78729852546547764433105024950, −13.33048366093382556056556432951, −11.79663371861190240268546608038, −10.56181131956372493637697962611, −9.525014035590283595941530276747, −8.902948268152799969935125871585, −7.32221313399423510103679923864, −5.93375983753768993663622318154, −4.68509123486423245817614273373, −2.11866516475300581575982482017,
0.61485910065012368066849884801, 2.13096528663464178911801423583, 4.10400780980834067458427286812, 6.58848621905720911150605483779, 7.45150931918547835744568724935, 8.687922094252393862126058229074, 10.05278762876120019975588758650, 10.88639382570577175546377269758, 12.15574088542854117599498218362, 13.20912277784373723631768930223