L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s − 5-s + 6-s + (0.367 + 1.13i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.867 − 2.66i)11-s + (−0.809 − 0.587i)12-s + (−2.01 + 1.46i)13-s + (0.367 − 1.13i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (1.51 − 4.67i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s − 0.447·5-s + 0.408·6-s + (0.138 + 0.427i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (−0.261 − 0.804i)11-s + (−0.233 − 0.169i)12-s + (−0.560 + 0.406i)13-s + (0.0981 − 0.302i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.368 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.857436 - 0.0432202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.857436 - 0.0432202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + (4.20 + 3.64i)T \) |
good | 7 | \( 1 + (-0.367 - 1.13i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (0.867 + 2.66i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (2.01 - 1.46i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.51 + 4.67i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.29 - 2.39i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.331 + 1.01i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.49 - 3.99i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + (-10.0 - 7.32i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-8.77 - 6.37i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (7.84 - 5.69i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.28 + 3.96i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.3 + 7.53i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.07T + 67T^{2} \) |
| 71 | \( 1 + (-4.35 + 13.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.98 - 6.11i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.85 - 14.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.5 - 8.36i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.27 + 6.98i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.377 + 1.16i)T + (-78.4 + 57.0i)T^{2} \) |
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
−9.917367149479644274063125805913, −9.431381047575721556672842907681, −8.455330226149661527140150011746, −7.67982829817117320576992498686, −6.76921577056702253895294509643, −5.61189542092179223127555139544, −4.76731278181208516729567315216, −3.55807323536578905207752834085, −2.57146079879295263548379184266, −0.824095338247814709825079315231,
0.816019940188614360441176854965, 2.28919546546848874497022329369, 3.89402289630719787317551954826, 5.01249526319257492874184846872, 5.80329606174161055626148521778, 7.04012530147674528268571558524, 7.39454630992595609868084694766, 8.226733276813987323872337070375, 9.192795751050995654395757491411, 10.28326746929108221228114452121