| L(s) = 1 | + (−23.2 − 29.1i)2-s + (−34.9 − 5.26i)3-s + (−195. + 854. i)4-s + (−829. − 565. i)5-s + (658. + 1.13e3i)6-s + (−4.56e3 + 7.90e3i)7-s + (1.22e4 − 5.89e3i)8-s + (−1.76e4 − 5.43e3i)9-s + (2.79e3 + 3.72e4i)10-s + (1.29e4 + 5.66e4i)11-s + (1.13e4 − 2.88e4i)12-s + (6.47e3 − 8.63e4i)13-s + (3.36e5 − 5.06e4i)14-s + (2.59e4 + 2.41e4i)15-s + (−5.16e4 − 2.48e4i)16-s + (−3.40e4 + 2.32e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.02 − 1.28i)2-s + (−0.248 − 0.0375i)3-s + (−0.380 + 1.66i)4-s + (−0.593 − 0.404i)5-s + (0.207 + 0.359i)6-s + (−0.718 + 1.24i)7-s + (1.05 − 0.508i)8-s + (−0.895 − 0.276i)9-s + (0.0883 + 1.17i)10-s + (0.266 + 1.16i)11-s + (0.157 − 0.401i)12-s + (0.0628 − 0.838i)13-s + (2.33 − 0.352i)14-s + (0.132 + 0.122i)15-s + (−0.197 − 0.0949i)16-s + (−0.0988 + 0.0674i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.228224 - 0.319480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228224 - 0.319480i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.16e7 - 5.78e6i)T \) |
| good | 2 | \( 1 + (23.2 + 29.1i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (34.9 + 5.26i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (829. + 565. i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (4.56e3 - 7.90e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-1.29e4 - 5.66e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (-6.47e3 + 8.63e4i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (3.40e4 - 2.32e4i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-5.05e5 + 1.55e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (1.36e6 - 1.26e6i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (7.09e6 - 1.06e6i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-1.20e6 + 3.08e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (1.56e6 + 2.71e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (1.83e6 + 2.30e6i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (4.90e6 - 2.15e7i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (7.01e6 + 9.36e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-1.47e8 - 7.11e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (2.18e7 + 5.55e7i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (-1.14e8 + 3.51e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (1.12e8 + 1.04e8i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (-1.88e6 + 2.51e7i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (-2.02e7 + 3.51e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.88e8 - 2.83e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (-1.08e9 - 1.63e8i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (2.99e8 + 1.31e9i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−12.84437609380850813810683201243, −12.06997537980379552526934569446, −11.41586533190518719618567767180, −9.802593526279528395179869679235, −9.061334524889510528484909609873, −7.83925980738253545810533786803, −5.68424641623398601045379543338, −3.49480985656356524607871154366, −2.17190110363678641081168929149, −0.37316955966672590441078198275,
0.61641603067025245027914560005, 3.64405360193386554981684320175, 5.82551246407818471233880469481, 6.89421054304939780229619254838, 7.907768963375674419316284381354, 9.144232901885450431708768766965, 10.46749561050597340761161560647, 11.56380521942831467515568835450, 13.72639151572495019307195759393, 14.48755305794680911323777446230
