| L(s) = 1 | − 3i·3-s − 20i·7-s − 9·9-s + 24·11-s − 74i·13-s + 54i·17-s − 124·19-s − 60·21-s − 120i·23-s + 27i·27-s + 78·29-s − 200·31-s − 72i·33-s − 70i·37-s − 222·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.07i·7-s − 0.333·9-s + 0.657·11-s − 1.57i·13-s + 0.770i·17-s − 1.49·19-s − 0.623·21-s − 1.08i·23-s + 0.192i·27-s + 0.499·29-s − 1.15·31-s − 0.379i·33-s − 0.311i·37-s − 0.911·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8920343187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8920343187\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 20iT - 343T^{2} \) |
| 11 | \( 1 - 24T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 54iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 124T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 78T + 2.43e4T^{2} \) |
| 31 | \( 1 + 200T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 330T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 24iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 450iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 24T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 - 196iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 288T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 520T + 4.93e5T^{2} \) |
| 83 | \( 1 - 156iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 286iT - 9.12e5T^{2} \) |
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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
−8.651835109093920234002452689501, −8.103260010753045588067054210085, −7.23430904923972942689520385288, −6.47395921539316655316216971336, −5.70177952596485951564210421818, −4.43074428845688217207928755300, −3.65130791461614091271262633170, −2.43084291535131809630774653369, −1.16956726109639021383296214670, −0.21732117931519244774116973146,
1.67509066677407199409440278082, 2.63722341833104839325747684010, 3.89548967951828831462284507502, 4.61159207576095690926885741894, 5.65462286231385025014553171046, 6.41331688706029227602565090476, 7.30557146623370923150234511029, 8.527548874405381277891954754166, 9.190427115456054174595655763355, 9.477869427899281601708240070049
