L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s + 12-s − 13-s + 4·14-s + 16-s − 17-s + 18-s − 2·19-s + 4·21-s + 24-s − 5·25-s − 26-s + 27-s + 4·28-s + 6·29-s + 8·31-s + 32-s − 34-s + 36-s + 2·37-s − 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.872·21-s + 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.975094384\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.975094384\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.46202664633796, −12.90400060855395, −12.29106093716310, −11.76379934829339, −11.63198380186366, −10.94797470913334, −10.49180905644518, −10.02829617429920, −9.478894630455590, −8.787184671333093, −8.291690543669222, −8.015779058718728, −7.530787585404195, −6.932934441742520, −6.375927181117979, −5.854975675747531, −5.193240774243223, −4.620002876938372, −4.449946768093457, −3.797208308433046, −3.120973704136600, −2.373233419574844, −2.141822820558764, −1.375259886545892, −0.7074342756939515,
0.7074342756939515, 1.375259886545892, 2.141822820558764, 2.373233419574844, 3.120973704136600, 3.797208308433046, 4.449946768093457, 4.620002876938372, 5.193240774243223, 5.854975675747531, 6.375927181117979, 6.932934441742520, 7.530787585404195, 8.015779058718728, 8.291690543669222, 8.787184671333093, 9.478894630455590, 10.02829617429920, 10.49180905644518, 10.94797470913334, 11.63198380186366, 11.76379934829339, 12.29106093716310, 12.90400060855395, 13.46202664633796