| L(s) = 1 | + (17.0 + 29.5i)2-s + (39.8 − 68.9i)3-s + (−327. + 567. i)4-s + (−711. − 1.23e3i)5-s + 2.72e3·6-s − 4.88e3·8-s + (6.66e3 + 1.15e4i)9-s + (2.43e4 − 4.21e4i)10-s + (−3.46e4 + 6.00e4i)11-s + (2.60e4 + 4.51e4i)12-s + 1.05e5·13-s − 1.13e5·15-s + (8.42e4 + 1.45e5i)16-s + (−2.84e5 + 4.92e5i)17-s + (−2.27e5 + 3.94e5i)18-s + (1.98e5 + 3.43e5i)19-s + ⋯ |
| L(s) = 1 | + (0.754 + 1.30i)2-s + (0.283 − 0.491i)3-s + (−0.639 + 1.10i)4-s + (−0.509 − 0.882i)5-s + 0.857·6-s − 0.421·8-s + (0.338 + 0.586i)9-s + (0.769 − 1.33i)10-s + (−0.714 + 1.23i)11-s + (0.363 + 0.629i)12-s + 1.02·13-s − 0.578·15-s + (0.321 + 0.556i)16-s + (−0.825 + 1.42i)17-s + (−0.511 + 0.886i)18-s + (0.348 + 0.604i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.23335 + 2.48796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23335 + 2.48796i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-17.0 - 29.5i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-39.8 + 68.9i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (711. + 1.23e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (3.46e4 - 6.00e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.05e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (2.84e5 - 4.92e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-1.98e5 - 3.43e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-3.10e5 - 5.37e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-7.12e5 + 1.23e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (6.55e6 + 1.13e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + 2.03e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.11e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-9.96e6 - 1.72e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (2.82e7 - 4.89e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-5.46e7 + 9.46e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.60e7 + 2.77e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (4.01e7 - 6.95e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.35e8 + 2.34e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.58e8 - 4.47e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 6.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-7.35e7 - 1.27e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.09e9T + 7.60e17T^{2} \) |
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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
−14.03329748840108218962445987681, −13.04523078729412179245170760023, −12.49060981300391389858178821810, −10.50851122107423550497141818651, −8.473249616811279413342663801854, −7.81187730863409569119016812437, −6.59097945495131835617201216559, −5.09673155892901584853614851869, −4.10636799490764550156487474770, −1.63392232448601354822554832690,
0.73741664302888416040555445476, 2.86726380761532578073847948618, 3.44687021367114543816156486067, 4.86542767385442711423346832045, 6.78991115341098259685186131151, 8.667801297812900528965287199211, 10.17640132040738175244874346312, 11.06760035785028997438517730294, 11.78666088616127497687909708275, 13.33095104115458877320075309340
