| L(s) = 1 | + (−0.970 + 0.239i)2-s + (1.21 + 1.07i)3-s + (0.885 − 0.464i)4-s + (2.48 + 3.60i)5-s + (−1.43 − 0.753i)6-s + (−2.49 − 2.21i)7-s + (−0.748 + 0.663i)8-s + (−0.0445 − 0.367i)9-s + (−3.27 − 2.90i)10-s + (−1.14 + 1.66i)11-s + (1.57 + 0.388i)12-s + (−0.434 + 0.227i)13-s + (2.95 + 1.54i)14-s + (−0.856 + 7.05i)15-s + (0.568 − 0.822i)16-s + (5.61 − 2.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.686 + 0.169i)2-s + (0.700 + 0.620i)3-s + (0.442 − 0.232i)4-s + (1.11 + 1.61i)5-s + (−0.586 − 0.307i)6-s + (−0.943 − 0.835i)7-s + (−0.264 + 0.234i)8-s + (−0.0148 − 0.122i)9-s + (−1.03 − 0.918i)10-s + (−0.346 + 0.502i)11-s + (0.454 + 0.112i)12-s + (−0.120 + 0.0632i)13-s + (0.789 + 0.414i)14-s + (−0.221 + 1.82i)15-s + (0.142 − 0.205i)16-s + (1.36 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.903952 + 0.624124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.903952 + 0.624124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.970 - 0.239i)T \) |
| 79 | \( 1 + (4.86 - 7.43i)T \) |
| good | 3 | \( 1 + (-1.21 - 1.07i)T + (0.361 + 2.97i)T^{2} \) |
| 5 | \( 1 + (-2.48 - 3.60i)T + (-1.77 + 4.67i)T^{2} \) |
| 7 | \( 1 + (2.49 + 2.21i)T + (0.843 + 6.94i)T^{2} \) |
| 11 | \( 1 + (1.14 - 1.66i)T + (-3.90 - 10.2i)T^{2} \) |
| 13 | \( 1 + (0.434 - 0.227i)T + (7.38 - 10.6i)T^{2} \) |
| 17 | \( 1 + (-5.61 + 2.94i)T + (9.65 - 13.9i)T^{2} \) |
| 19 | \( 1 + (-0.742 - 1.95i)T + (-14.2 + 12.5i)T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + (-0.993 + 8.18i)T + (-28.1 - 6.94i)T^{2} \) |
| 31 | \( 1 + (-4.21 + 1.03i)T + (27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (0.505 + 1.33i)T + (-27.6 + 24.5i)T^{2} \) |
| 41 | \( 1 + (-3.20 - 4.63i)T + (-14.5 + 38.3i)T^{2} \) |
| 43 | \( 1 + (6.75 + 9.78i)T + (-15.2 + 40.2i)T^{2} \) |
| 47 | \( 1 + (-1.02 + 2.71i)T + (-35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (4.53 - 4.01i)T + (6.38 - 52.6i)T^{2} \) |
| 59 | \( 1 + (-3.42 - 1.79i)T + (33.5 + 48.5i)T^{2} \) |
| 61 | \( 1 + (1.07 + 2.82i)T + (-45.6 + 40.4i)T^{2} \) |
| 67 | \( 1 + (-0.801 - 0.197i)T + (59.3 + 31.1i)T^{2} \) |
| 71 | \( 1 + (2.43 - 2.15i)T + (8.55 - 70.4i)T^{2} \) |
| 73 | \( 1 + (2.64 + 1.38i)T + (41.4 + 60.0i)T^{2} \) |
| 83 | \( 1 + (-3.94 + 2.07i)T + (47.1 - 68.3i)T^{2} \) |
| 89 | \( 1 + (-8.34 - 7.39i)T + (10.7 + 88.3i)T^{2} \) |
| 97 | \( 1 + (4.90 + 12.9i)T + (-72.6 + 64.3i)T^{2} \) |
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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
−13.58269043995357744495210858098, −11.90448237522021642389049638146, −10.40988456379271229714271545552, −9.897825334350460267227374532758, −9.629362701623274824711537592446, −7.81914248936520496865456725734, −6.84934991592770049615788647339, −5.92709083224580425232587569176, −3.60117581910570261016648716498, −2.56534857474652244772813693773,
1.51482872337268492686718164585, 2.88183031977690565553212264742, 5.27815568085422365256841596896, 6.25130862195931532018112293903, 7.989418776555384520838511404949, 8.643163220985388216990021029002, 9.450721334572874217371892578577, 10.28130288594044655976639409122, 12.14718187595290344598579069470, 12.72518274293419629681933759205
