L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s + 15-s − 16-s − 2·17-s − 18-s − 4·19-s + 20-s − 4·22-s − 3·24-s + 25-s + 2·26-s − 27-s − 2·29-s − 30-s − 5·32-s − 4·33-s + 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.182·30-s − 0.883·32-s − 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6656790232\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6656790232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.75608132970451643863162535345, −7.35063812070988934629687926389, −6.50280165222571433827222186827, −5.90152994471802728386246983536, −4.87601044783691669506154377205, −4.28778561739561775104237955771, −3.84287119870099927831381708651, −2.48628619288857145159052937052, −1.41588766872511613687204474026, −0.50790911508031838973139290532,
0.50790911508031838973139290532, 1.41588766872511613687204474026, 2.48628619288857145159052937052, 3.84287119870099927831381708651, 4.28778561739561775104237955771, 4.87601044783691669506154377205, 5.90152994471802728386246983536, 6.50280165222571433827222186827, 7.35063812070988934629687926389, 7.75608132970451643863162535345