L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.206 + 0.770i)3-s + (0.866 + 0.499i)4-s + (−0.398 + 0.690i)6-s + (0.349 + 0.349i)7-s + (0.707 + 0.707i)8-s + (2.04 + 1.18i)9-s − 1.21·11-s + (−0.563 + 0.563i)12-s + (6.55 − 1.75i)13-s + (0.246 + 0.427i)14-s + (0.500 + 0.866i)16-s + (1.57 − 5.86i)17-s + (1.67 + 1.67i)18-s + (−4.09 − 1.48i)19-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.119 + 0.444i)3-s + (0.433 + 0.249i)4-s + (−0.162 + 0.281i)6-s + (0.131 + 0.131i)7-s + (0.249 + 0.249i)8-s + (0.682 + 0.394i)9-s − 0.367·11-s + (−0.162 + 0.162i)12-s + (1.81 − 0.487i)13-s + (0.0659 + 0.114i)14-s + (0.125 + 0.216i)16-s + (0.381 − 1.42i)17-s + (0.394 + 0.394i)18-s + (−0.940 − 0.340i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.25450 + 1.22264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.25450 + 1.22264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.09 + 1.48i)T \) |
good | 3 | \( 1 + (0.206 - 0.770i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (-0.349 - 0.349i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 13 | \( 1 + (-6.55 + 1.75i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 5.86i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-2.37 - 8.86i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (3.16 - 5.47i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.84iT - 31T^{2} \) |
| 37 | \( 1 + (2.41 - 2.41i)T - 37iT^{2} \) |
| 41 | \( 1 + (3.55 - 2.05i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.62 - 1.50i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (0.201 - 0.0539i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-13.1 + 3.51i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.0144 + 0.0250i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.90 + 6.75i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.56 + 13.3i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (5.99 - 3.46i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.53 + 2.01i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.17 - 3.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.36 + 2.36i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.10 - 3.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.06 + 0.821i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.38642104730398422933966348038, −9.340224004128458365448505074475, −8.471261969149745793312358172348, −7.48907079898401666899408017018, −6.75091256778235317756957231956, −5.51947733958452600455997195982, −5.08207029254624050772826589652, −3.90208336041362916376097874321, −3.11199124293841901492168410246, −1.53461405144800349230245127806,
1.17154939274055458420407406220, 2.29229939617469465595576130203, 3.93410094289633992291313863322, 4.18807617391475059862150086817, 5.86099353768949925272438667184, 6.23728343101137387059297501345, 7.17474504759322269253853194459, 8.234270611537970631751717236671, 8.928046938182989087206924188721, 10.36752112750729622184334423351