L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (2 − 3.46i)5-s + (0.499 + 0.866i)6-s − 3·7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.99 + 3.46i)10-s + 2·11-s − 0.999·12-s + (3.5 + 6.06i)13-s + (1.5 − 2.59i)14-s + (−1.99 − 3.46i)15-s + (−0.5 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.894 − 1.54i)5-s + (0.204 + 0.353i)6-s − 1.13·7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.632 + 1.09i)10-s + 0.603·11-s − 0.288·12-s + (0.970 + 1.68i)13-s + (0.400 − 0.694i)14-s + (−0.516 − 0.894i)15-s + (−0.125 + 0.216i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.967227 - 0.209044i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.967227 - 0.209044i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (4 - 1.73i)T \) |
good | 5 | \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-3.5 - 6.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1 - 1.73i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 - 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.37268083446081823670774685786, −12.93675058068021834249048812864, −11.64041903378439221298354564433, −9.698975745534639055029712035201, −9.197588431634538098208427051470, −8.356139788180140540496179767920, −6.62925058745666810014651140038, −5.95861093515340541713444358495, −4.25104104797604112613275752221, −1.54913072358068928721326785821,
2.70076862898533993905049973728, 3.56266624530711247778126453860, 5.91093817892060880743479689853, 6.93071666202233534660996397495, 8.609816077197101248862353423767, 9.759312975536325898130913456397, 10.44568889777340241685417550840, 11.07956292177406139206527796372, 12.81662285296933955415682074037, 13.52075601597673538336852622496