| L(s) = 1 | − 25.7·2-s + (66.1 − 114. i)3-s + 151.·4-s + (−107. + 185. i)5-s + (−1.70e3 + 2.95e3i)6-s + (−2.35e3 − 4.08e3i)7-s + 9.28e3·8-s + (1.08e3 + 1.88e3i)9-s + (2.76e3 − 4.78e3i)10-s + 4.06e4·11-s + (1.00e4 − 1.73e4i)12-s + (2.20e4 + 3.81e4i)13-s + (6.07e4 + 1.05e5i)14-s + (1.41e4 + 2.45e4i)15-s − 3.16e5·16-s + (1.42e5 + 2.47e5i)17-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + (0.471 − 0.816i)3-s + 0.296·4-s + (−0.0766 + 0.132i)5-s + (−0.536 + 0.929i)6-s + (−0.371 − 0.643i)7-s + 0.801·8-s + (0.0552 + 0.0957i)9-s + (0.0872 − 0.151i)10-s + 0.836·11-s + (0.139 − 0.241i)12-s + (0.213 + 0.370i)13-s + (0.422 + 0.732i)14-s + (0.0723 + 0.125i)15-s − 1.20·16-s + (0.414 + 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.891875 - 0.678918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891875 - 0.678918i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.90e7 + 1.18e7i)T \) |
| good | 2 | \( 1 + 25.7T + 512T^{2} \) |
| 3 | \( 1 + (-66.1 + 114. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (107. - 185. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 + (2.35e3 + 4.08e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 4.06e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-2.20e4 - 3.81e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.42e5 - 2.47e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (2.90e5 - 5.02e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.10e6 + 1.90e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.97e6 + 5.15e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + (-2.58e6 + 4.47e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.38e6 + 1.45e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 3.01e6T + 3.27e14T^{2} \) |
| 47 | \( 1 + 1.15e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (2.44e7 - 4.23e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 - 1.15e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + (1.84e7 + 3.19e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-9.56e7 + 1.65e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.35e7 + 1.10e8i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-2.42e7 - 4.20e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-6.59e7 - 1.14e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (1.42e8 - 2.45e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (3.45e8 - 5.98e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 9.14e8T + 7.60e17T^{2} \) |
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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
−13.65092905232821996634974888538, −12.70678861966392144499509963872, −11.00332674877184089240742353044, −9.885854795754599609292429095404, −8.634916598275081760107731242898, −7.64617623335788728028498707638, −6.58606731787442176700353864504, −4.06135138835557267241773073588, −1.91547181530730752861972354465, −0.71100653726445187182790046397,
1.07882422223841693412836061302, 3.19431654783646599520843037370, 4.82156731762803517028748976643, 6.89505465237936897706873369906, 8.578310552573234810619037811454, 9.232993034623391152805637509478, 10.08929481556035961037011515184, 11.48690691923808813886294571514, 13.04547099476039225966453782488, 14.51157098658462913061592942784
