L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 5·7-s − 8-s + 9-s + 4·10-s − 4·11-s − 12-s − 5·13-s + 5·14-s + 4·15-s + 16-s − 6·17-s − 18-s − 7·19-s − 4·20-s + 5·21-s + 4·22-s − 4·23-s + 24-s + 11·25-s + 5·26-s − 27-s − 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.288·12-s − 1.38·13-s + 1.33·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.60·19-s − 0.894·20-s + 1.09·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 11/5·25-s + 0.980·26-s − 0.192·27-s − 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 3643 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−16.22541925198463, −15.98544034315667, −15.51521756245692, −14.99701989082795, −14.55593186446272, −13.14503942312672, −12.90840896000810, −12.63070665557572, −11.95582133893510, −11.33132275124065, −10.92484853233004, −10.21890907222734, −9.939382880677538, −9.098930739365421, −8.576732473945906, −7.897289349700507, −7.343606725321433, −6.820269385466602, −6.477367035322905, −5.568706416124283, −4.735389080146614, −4.121460403163850, −3.442506884171130, −2.735744438787194, −2.007201679006132, 0, 0, 0,
2.007201679006132, 2.735744438787194, 3.442506884171130, 4.121460403163850, 4.735389080146614, 5.568706416124283, 6.477367035322905, 6.820269385466602, 7.343606725321433, 7.897289349700507, 8.576732473945906, 9.098930739365421, 9.939382880677538, 10.21890907222734, 10.92484853233004, 11.33132275124065, 11.95582133893510, 12.63070665557572, 12.90840896000810, 13.14503942312672, 14.55593186446272, 14.99701989082795, 15.51521756245692, 15.98544034315667, 16.22541925198463