L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 3·11-s + 12-s + 13-s − 2·14-s + 16-s − 17-s − 18-s + 2·19-s + 2·21-s − 3·22-s − 24-s − 26-s + 27-s + 2·28-s − 6·29-s − 4·31-s − 32-s + 3·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s − 0.639·22-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.605594046\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.605594046\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.84233179646202, −14.59328097367259, −14.18065615544873, −13.41537108726885, −13.00142512277432, −12.24562853743961, −11.71331966948326, −11.26614708174387, −10.74891136003049, −10.15012381433276, −9.469669144300448, −9.003143873941292, −8.720833881537850, −7.916438277631627, −7.546061437113742, −6.978295870627811, −6.329922654361043, −5.616617636722468, −4.996852945056612, −4.061450889640150, −3.713043946776161, −2.826864666805875, −2.015626446182528, −1.520366161731250, −0.6807789178533634,
0.6807789178533634, 1.520366161731250, 2.015626446182528, 2.826864666805875, 3.713043946776161, 4.061450889640150, 4.996852945056612, 5.616617636722468, 6.329922654361043, 6.978295870627811, 7.546061437113742, 7.916438277631627, 8.720833881537850, 9.003143873941292, 9.469669144300448, 10.15012381433276, 10.74891136003049, 11.26614708174387, 11.71331966948326, 12.24562853743961, 13.00142512277432, 13.41537108726885, 14.18065615544873, 14.59328097367259, 14.84233179646202