| L(s) = 1 | + (25.7 + 44.6i)2-s + (−329. − 261. i)3-s + (−302. + 523. i)4-s + (4.46e3 − 7.73e3i)5-s + (3.20e3 − 2.14e4i)6-s + (−5.87e3 − 1.01e4i)7-s + 7.43e4·8-s + (3.98e4 + 1.72e5i)9-s + 4.59e5·10-s + (−3.42e5 − 5.93e5i)11-s + (2.36e5 − 9.32e4i)12-s + (9.24e5 − 1.60e6i)13-s + (3.02e5 − 5.24e5i)14-s + (−3.49e6 + 1.37e6i)15-s + (2.53e6 + 4.38e6i)16-s − 2.62e6·17-s + ⋯ |
| L(s) = 1 | + (0.569 + 0.985i)2-s + (−0.782 − 0.622i)3-s + (−0.147 + 0.255i)4-s + (0.638 − 1.10i)5-s + (0.168 − 1.12i)6-s + (−0.132 − 0.228i)7-s + 0.802·8-s + (0.225 + 0.974i)9-s + 1.45·10-s + (−0.642 − 1.11i)11-s + (0.274 − 0.108i)12-s + (0.690 − 1.19i)13-s + (0.150 − 0.260i)14-s + (−1.18 + 0.468i)15-s + (0.604 + 1.04i)16-s − 0.448·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.76700 - 0.591034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76700 - 0.591034i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (329. + 261. i)T \) |
| good | 2 | \( 1 + (-25.7 - 44.6i)T + (-1.02e3 + 1.77e3i)T^{2} \) |
| 5 | \( 1 + (-4.46e3 + 7.73e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 7 | \( 1 + (5.87e3 + 1.01e4i)T + (-9.88e8 + 1.71e9i)T^{2} \) |
| 11 | \( 1 + (3.42e5 + 5.93e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-9.24e5 + 1.60e6i)T + (-8.96e11 - 1.55e12i)T^{2} \) |
| 17 | \( 1 + 2.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 3.45e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (1.51e7 - 2.61e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-1.10e8 - 1.90e8i)T + (-6.10e15 + 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-7.20e7 + 1.24e8i)T + (-1.27e16 - 2.20e16i)T^{2} \) |
| 37 | \( 1 - 1.39e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (3.21e7 - 5.57e7i)T + (-2.75e17 - 4.76e17i)T^{2} \) |
| 43 | \( 1 + (-8.71e8 - 1.50e9i)T + (-4.64e17 + 8.04e17i)T^{2} \) |
| 47 | \( 1 + (5.85e8 + 1.01e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + 1.46e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (3.43e9 - 5.95e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-5.11e8 - 8.85e8i)T + (-2.17e19 + 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-6.27e9 + 1.08e10i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.72e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.96e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (9.17e9 + 1.58e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-7.19e9 - 1.24e10i)T + (-6.43e20 + 1.11e21i)T^{2} \) |
| 89 | \( 1 - 1.45e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-4.14e10 - 7.18e10i)T + (-3.57e21 + 6.19e21i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.96740996340512801849950672825, −16.71408362284747577930979424506, −15.85320427698484429062620920043, −13.62334001882345928764587015283, −12.94777606998929651986659019729, −10.72308346048581797995639075117, −8.094680269866507920476252638445, −6.13358709949497523835957857526, −5.16472180431327338126107285685, −1.01874878581330989522486189536,
2.34560145713543696230174403123, 4.34847427730169564036624561105, 6.53569460004065468040103193778, 9.981612824006160188341356400930, 10.97770239382957547508434867897, 12.31869784107376887495855391754, 14.02860023192474558997949372466, 15.72622678036557788348525386575, 17.45580184555587961849896975979, 18.76282261709354270794785378982
