L(s) = 1 | + (0.818 − 2.08i)5-s + (0.623 + 0.623i)7-s − 1.13i·11-s + (1.31 − 1.31i)13-s + (−4.45 + 4.45i)17-s − 0.624i·19-s + (−0.707 − 0.707i)23-s + (−3.65 − 3.40i)25-s + 3.20·29-s + 9.14·31-s + (1.80 − 0.787i)35-s + (−0.268 − 0.268i)37-s − 1.26i·41-s + (7.06 − 7.06i)43-s + (1.93 − 1.93i)47-s + ⋯ |
L(s) = 1 | + (0.366 − 0.930i)5-s + (0.235 + 0.235i)7-s − 0.341i·11-s + (0.364 − 0.364i)13-s + (−1.08 + 1.08i)17-s − 0.143i·19-s + (−0.147 − 0.147i)23-s + (−0.731 − 0.681i)25-s + 0.595·29-s + 1.64·31-s + (0.305 − 0.133i)35-s + (−0.0441 − 0.0441i)37-s − 0.197i·41-s + (1.07 − 1.07i)43-s + (0.282 − 0.282i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00877 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00877 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.829091539\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.829091539\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.818 + 2.08i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-0.623 - 0.623i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.13iT - 11T^{2} \) |
| 13 | \( 1 + (-1.31 + 1.31i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.45 - 4.45i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.624iT - 19T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 9.14T + 31T^{2} \) |
| 37 | \( 1 + (0.268 + 0.268i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.26iT - 41T^{2} \) |
| 43 | \( 1 + (-7.06 + 7.06i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.93 + 1.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.62 + 1.62i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 - 0.362T + 61T^{2} \) |
| 67 | \( 1 + (4.96 + 4.96i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (-7.41 + 7.41i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.11iT - 79T^{2} \) |
| 83 | \( 1 + (6.52 + 6.52i)T + 83iT^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + (-1.67 - 1.67i)T + 97iT^{2} \) |
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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
−8.400815283671215647971522683640, −7.73372464029006665572775312005, −6.51574238431435077843778547113, −6.08899954298005757485991676647, −5.21887723946893398364007505839, −4.52673585700859685021603185636, −3.75719560779999038335872252413, −2.55299468405457497016543389239, −1.68021507559857885141717332579, −0.55138481234884554342858282709,
1.18475877240634149914064584635, 2.39651641325897808773935677007, 2.94810967825106207156958912073, 4.16343160325461367807761787457, 4.70658609723782943272668873512, 5.80609703843340279027212003526, 6.50424408207834472829030285655, 7.03708907314668404490065228389, 7.77934528663965570150732842373, 8.568334434543514079186825703500