L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 2·11-s + 12-s − 13-s + 15-s + 16-s − 18-s − 2·19-s + 20-s + 2·22-s − 24-s − 4·25-s + 26-s + 27-s − 29-s − 30-s − 8·31-s − 32-s − 2·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.426·22-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.185·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155526 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9800079155\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9800079155\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.24317358769899, −12.86802735855493, −12.38412343507290, −11.92514356482999, −11.22042137171599, −10.69231990648442, −10.54203444559045, −9.738718832066488, −9.519562846668032, −9.008497010981425, −8.565345259592299, −7.814855080654327, −7.771482848867525, −7.003300608999515, −6.650712842395811, −5.850039889839879, −5.533028673053257, −4.889718061054864, −4.151541150581177, −3.614590695461657, −2.966761787729175, −2.359935352298850, −1.903444181542880, −1.346597314122448, −0.2988624933831015,
0.2988624933831015, 1.346597314122448, 1.903444181542880, 2.359935352298850, 2.966761787729175, 3.614590695461657, 4.151541150581177, 4.889718061054864, 5.533028673053257, 5.850039889839879, 6.650712842395811, 7.003300608999515, 7.771482848867525, 7.814855080654327, 8.565345259592299, 9.008497010981425, 9.519562846668032, 9.738718832066488, 10.54203444559045, 10.69231990648442, 11.22042137171599, 11.92514356482999, 12.38412343507290, 12.86802735855493, 13.24317358769899