| L(s) = 1 | + (−6.47 − 19.9i)2-s + (−21.8 − 15.8i)3-s + (−251. + 182. i)4-s + (−11.5 + 35.6i)5-s + (−174. + 537. i)6-s + (80.4 − 58.4i)7-s + (3.08e3 + 2.24e3i)8-s + (225. + 693. i)9-s + 784.·10-s + (−1.38e3 + 4.19e3i)11-s + 8.38e3·12-s + (−2.82e3 − 8.70e3i)13-s + (−1.68e3 − 1.22e3i)14-s + (818. − 594. i)15-s + (1.24e4 − 3.82e4i)16-s + (−3.56e3 + 1.09e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.571 − 1.76i)2-s + (−0.467 − 0.339i)3-s + (−1.96 + 1.42i)4-s + (−0.0414 + 0.127i)5-s + (−0.330 + 1.01i)6-s + (0.0886 − 0.0644i)7-s + (2.13 + 1.55i)8-s + (0.103 + 0.317i)9-s + 0.248·10-s + (−0.313 + 0.949i)11-s + 1.40·12-s + (−0.356 − 1.09i)13-s + (−0.164 − 0.119i)14-s + (0.0626 − 0.0455i)15-s + (0.758 − 2.33i)16-s + (−0.175 + 0.541i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.542389 - 0.100132i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.542389 - 0.100132i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (21.8 + 15.8i)T \) |
| 11 | \( 1 + (1.38e3 - 4.19e3i)T \) |
| good | 2 | \( 1 + (6.47 + 19.9i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (11.5 - 35.6i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-80.4 + 58.4i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (2.82e3 + 8.70e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (3.56e3 - 1.09e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.00e4 + 7.32e3i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 9.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.83e5 - 1.33e5i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-5.58e4 - 1.71e5i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.47e5 + 1.06e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-5.08e5 - 3.69e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 5.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.76e5 - 1.28e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-1.26e5 - 3.90e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.16e6 - 1.57e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.87e5 + 8.84e5i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 + 2.81e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-3.04e5 + 9.38e5i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (3.48e6 - 2.53e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (2.42e6 + 7.46e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (1.98e6 - 6.12e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + 4.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.39e6 - 4.29e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.92580045516812890762084023320, −12.96101755784721242443813994801, −12.64068055567550196536321390249, −11.13503948496598417410717536389, −10.43968709712883854883643259738, −9.056267770363698050681785393784, −7.48747398146341165752836681590, −4.85044571126495049827572277032, −2.92091917111898023666679269815, −1.28579085726404594070477670592,
0.37694644743608404559776429542, 4.62084838305512062643840615074, 5.90046704646414259813137899218, 7.13824524068773026153516362596, 8.600431308006056781557155483911, 9.625489587091007301622478769793, 11.24256509831030487572658745650, 13.29902275795302543828215138354, 14.52108090471012134347812534156, 15.49779956351427200127755933178
