| L(s) = 1 | + (−0.5 + 0.866i)2-s + (7.5 + 12.9i)4-s − 30.9·8-s + (−40.5 + 70.1i)9-s + (103 + 178. i)11-s + (−104. + 180. i)16-s + (−40.5 − 70.1i)18-s − 206·22-s + (367 − 635. i)23-s + (−312.5 − 541. i)25-s + 1.23e3·29-s + (−352. − 610. i)32-s − 1.21e3·36-s + (647 − 1.12e3i)37-s − 334·43-s + (−1.54e3 + 2.67e3i)44-s + ⋯ |
| L(s) = 1 | + (−0.125 + 0.216i)2-s + (0.468 + 0.811i)4-s − 0.484·8-s + (−0.5 + 0.866i)9-s + (0.851 + 1.47i)11-s + (−0.408 + 0.707i)16-s + (−0.125 − 0.216i)18-s − 0.425·22-s + (0.693 − 1.20i)23-s + (−0.5 − 0.866i)25-s + 1.46·29-s + (−0.344 − 0.596i)32-s − 0.937·36-s + (0.472 − 0.818i)37-s − 0.180·43-s + (−0.798 + 1.38i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.993134 + 1.05815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.993134 + 1.05815i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (0.5 - 0.866i)T + (-8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (-103 - 178. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-367 + 635. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 1.23e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-647 + 1.12e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 + 334T + 3.41e6T^{2} \) |
| 47 | \( 1 + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-2.79e3 - 4.83e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.47e3 + 4.28e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 2.91e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.82e3 + 3.15e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 8.85e7T^{2} \) |
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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
−15.24961562831167636606685899898, −14.10440286114921863442859588217, −12.64505158838501834716532760603, −11.82661709222005691312348018295, −10.47007656057678936234828793449, −8.887337718285025636834792151351, −7.67744387118201432369994400349, −6.50657966749115155852874810212, −4.45863280702003633656854553125, −2.43063663150742546720321985454,
0.987027341325274920568579000502, 3.27881667833585164453089653318, 5.63657384310397046760815961931, 6.67524590754859883805106535306, 8.676759312273618569918526911138, 9.737257815540610610359955395362, 11.21822405260010153483571339005, 11.80032090769131054027101039065, 13.60382604080703148038476002507, 14.60144560973149915620499630153
