L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.266 − 0.820i)3-s + (0.309 − 0.951i)4-s + (1.93 − 1.11i)5-s + (0.697 + 0.507i)6-s + (−2.50 + 0.853i)7-s + (0.309 + 0.951i)8-s + (1.82 − 1.32i)9-s + (−0.908 + 2.04i)10-s + (−3.27 − 0.526i)11-s − 0.862·12-s + (−1.64 − 2.26i)13-s + (1.52 − 2.16i)14-s + (−1.43 − 1.28i)15-s + (−0.809 − 0.587i)16-s + (1.88 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.153 − 0.473i)3-s + (0.154 − 0.475i)4-s + (0.865 − 0.500i)5-s + (0.284 + 0.206i)6-s + (−0.946 + 0.322i)7-s + (0.109 + 0.336i)8-s + (0.608 − 0.442i)9-s + (−0.287 + 0.646i)10-s + (−0.987 − 0.158i)11-s − 0.249·12-s + (−0.455 − 0.627i)13-s + (0.407 − 0.577i)14-s + (−0.370 − 0.333i)15-s + (−0.202 − 0.146i)16-s + (0.456 − 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.383458 - 0.641813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.383458 - 0.641813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (2.50 - 0.853i)T \) |
| 11 | \( 1 + (3.27 + 0.526i)T \) |
good | 3 | \( 1 + (0.266 + 0.820i)T + (-2.42 + 1.76i)T^{2} \) |
| 13 | \( 1 + (1.64 + 2.26i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.88 + 2.59i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.715 + 2.20i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 5.07iT - 23T^{2} \) |
| 29 | \( 1 + (4.94 + 1.60i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.37 - 3.27i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (4.11 + 1.33i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.541 + 1.66i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 9.38T + 43T^{2} \) |
| 47 | \( 1 + (2.33 + 7.17i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.07 + 1.48i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-9.20 - 2.98i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (6.40 + 4.64i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 7.62iT - 67T^{2} \) |
| 71 | \( 1 + (6.31 + 4.58i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-5.53 - 1.79i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.48 - 10.2i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.00 + 6.88i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + (2.05 - 1.49i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.826143305513909419912371370727, −9.355879002490204059094401609342, −8.335448263135996985181042639472, −7.34971346603830002958336719331, −6.64259409254071747460383975693, −5.67037919033428041063668420320, −5.08159302617028912137251620420, −3.23464987142471898161869606605, −1.96258220379480184570259084361, −0.44057965993894272043735998049,
1.81908316974579378328670092831, 2.89716184107922755807893821333, 4.05431124707617168399604185333, 5.23274888168989778861934057409, 6.35485587902783851080550348511, 7.14488453596193965737790759710, 8.077385030833681541645262365782, 9.299419690603646814460383874336, 9.972430321207607764041371345387, 10.34288384805069904549776530141