L(s) = 1 | + (−0.623 + 0.781i)3-s + (0.900 + 0.433i)4-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)12-s + (−0.623 − 0.781i)13-s + (0.623 + 0.781i)16-s + 1.94i·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.623 + 0.781i)28-s + 0.867i·31-s + (0.222 − 0.974i)36-s + (−0.376 − 0.781i)37-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)3-s + (0.900 + 0.433i)4-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (−0.900 + 0.433i)12-s + (−0.623 − 0.781i)13-s + (0.623 + 0.781i)16-s + 1.94i·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.623 + 0.781i)28-s + 0.867i·31-s + (0.222 − 0.974i)36-s + (−0.376 − 0.781i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.005725854\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.005725854\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.623 - 0.781i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.623 + 0.781i)T \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 5 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 - 1.94iT - T^{2} \) |
| 23 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 - 0.867iT - T^{2} \) |
| 37 | \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 61 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 + 1.56iT - T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.846 + 0.193i)T + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 - 1.80T + T^{2} \) |
| 83 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 + 1.94iT - T^{2} \) |
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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
−9.774239809599026498905677003489, −8.935654305116383632861020210036, −8.088449996320534244176818232824, −7.27543412555652493635199551259, −6.26164173870512164441170030193, −5.72093031945020412437277812806, −4.98743150929421801605149405899, −3.63322635866381646480742901507, −3.08759018173970382560289987292, −1.81831284451908994263148675804,
0.77118392340536878625249747484, 2.02250076147576288551378622546, 2.89633208117426421068227496961, 4.44472019082079479727064638987, 5.13472805347537901958332859602, 6.33416855029500862830533928661, 6.71916980543078852754517778165, 7.30556198990110897466924449463, 8.046403408430635614730336431615, 9.275750490883187339939662660506