L(s) = 1 | + (−1 + i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (−0.435 − 4.98i)5-s − 2.44·6-s + (6.78 − 6.78i)7-s + (2 + 2i)8-s + 2.99i·9-s + (5.41 + 4.54i)10-s + 9.68·11-s + (2.44 − 2.44i)12-s + (−12.5 − 12.5i)13-s + 13.5i·14-s + (5.56 − 6.63i)15-s − 4·16-s + (−2.02 + 2.02i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (−0.0871 − 0.996i)5-s − 0.408·6-s + (0.969 − 0.969i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.541 + 0.454i)10-s + 0.880·11-s + (0.204 − 0.204i)12-s + (−0.965 − 0.965i)13-s + 0.969i·14-s + (0.371 − 0.442i)15-s − 0.250·16-s + (−0.119 + 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.465858384\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.465858384\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 + (0.435 + 4.98i)T \) |
| 23 | \( 1 + (-3.39 - 3.39i)T \) |
good | 7 | \( 1 + (-6.78 + 6.78i)T - 49iT^{2} \) |
| 11 | \( 1 - 9.68T + 121T^{2} \) |
| 13 | \( 1 + (12.5 + 12.5i)T + 169iT^{2} \) |
| 17 | \( 1 + (2.02 - 2.02i)T - 289iT^{2} \) |
| 19 | \( 1 + 5.79iT - 361T^{2} \) |
| 29 | \( 1 + 25.2iT - 841T^{2} \) |
| 31 | \( 1 + 15.3T + 961T^{2} \) |
| 37 | \( 1 + (25.5 - 25.5i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 31.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (15.7 + 15.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-52.4 + 52.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (38.7 + 38.7i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 9.40iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 109.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-67.8 + 67.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 112.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-44.3 - 44.3i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 25.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-101. - 101. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 43.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-88.0 + 88.0i)T - 9.40e3iT^{2} \) |
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
−9.901463205134393551062614187491, −9.178467948416680162054953451481, −8.255410310068294951059830769983, −7.77368867100232303367376338620, −6.79902389479563335394403352427, −5.33314566133773362201457178392, −4.72313717327438907365282352027, −3.72820466673873089418949126976, −1.84402401522144560107363938738, −0.58804164487083887607697104792,
1.67981298405824100276007437738, 2.41560807022884988482277738475, 3.57773082000378903275564073917, 4.83003225640231380889208106871, 6.24344366992063591554289713913, 7.13934962479414972738942421158, 7.84297875799002141802814324410, 8.929425926193100810717073407178, 9.344743400924781059426170335997, 10.52210210768609303316901435724