| L(s) = 1 | + (−1.52 + 0.449i)2-s + (−0.654 − 0.755i)3-s + (0.455 − 0.293i)4-s + (0.142 + 0.989i)5-s + (1.34 + 0.861i)6-s + (0.415 − 0.909i)7-s + (1.52 − 1.75i)8-s + (−0.142 + 0.989i)9-s + (−0.662 − 1.45i)10-s + (−2.84 − 0.835i)11-s + (−0.520 − 0.152i)12-s + (1.10 + 2.42i)13-s + (−0.226 + 1.57i)14-s + (0.654 − 0.755i)15-s + (−1.98 + 4.35i)16-s + (−4.41 − 2.83i)17-s + ⋯ |
| L(s) = 1 | + (−1.08 + 0.317i)2-s + (−0.378 − 0.436i)3-s + (0.227 − 0.146i)4-s + (0.0636 + 0.442i)5-s + (0.547 + 0.351i)6-s + (0.157 − 0.343i)7-s + (0.538 − 0.621i)8-s + (−0.0474 + 0.329i)9-s + (−0.209 − 0.458i)10-s + (−0.858 − 0.252i)11-s + (−0.150 − 0.0440i)12-s + (0.307 + 0.673i)13-s + (−0.0606 + 0.421i)14-s + (0.169 − 0.195i)15-s + (−0.497 + 1.08i)16-s + (−1.06 − 0.687i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.426513 - 0.270586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.426513 - 0.270586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-3.73 - 3.01i)T \) |
| good | 2 | \( 1 + (1.52 - 0.449i)T + (1.68 - 1.08i)T^{2} \) |
| 5 | \( 1 + (-0.142 - 0.989i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (2.84 + 0.835i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.10 - 2.42i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (4.41 + 2.83i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 1.20i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (6.05 + 3.89i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-6.01 + 6.93i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 10.0i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.04 + 7.26i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.45 + 5.13i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 + (0.585 - 1.28i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.58 + 10.0i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.11 - 2.43i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-13.2 + 3.90i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-11.2 + 3.30i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.02 - 2.58i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.30 - 13.8i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.41 - 9.82i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (3.36 + 3.88i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.48 - 10.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.03320671723607673736754831659, −9.761303661401779798026185753547, −9.041718437088745302996437400160, −8.024743365551516290514350491348, −7.24046744356225422353349804811, −6.63922327348981965014025030632, −5.30665705004671012113511969512, −4.02014711815903424442156554349, −2.28544615703709038671657901214, −0.51973339064833922302058213659,
1.27890950361758601942651046605, 2.88815469962040436057406342996, 4.70184675787219996747663505297, 5.26104687067170595216948567884, 6.61218604452750462707584118115, 7.971206842031117985953417998665, 8.604977250282441766593570271392, 9.353022438552112417494760672072, 10.36277394987317576561788522252, 10.76020778759524270750095055472
