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
−9.477869427899281601708240070049, −9.190427115456054174595655763355, −8.527548874405381277891954754166, −7.30557146623370923150234511029, −6.41331688706029227602565090476, −5.65462286231385025014553171046, −4.61159207576095690926885741894, −3.89548967951828831462284507502, −2.63722341833104839325747684010, −1.67509066677407199409440278082,
0.21732117931519244774116973146, 1.16956726109639021383296214670, 2.43084291535131809630774653369, 3.65130791461614091271262633170, 4.43074428845688217207928755300, 5.70177952596485951564210421818, 6.47395921539316655316216971336, 7.23430904923972942689520385288, 8.103260010753045588067054210085, 8.651835109093920234002452689501