L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s − 2·13-s − 4·14-s − 15-s + 16-s + 18-s − 19-s + 20-s + 4·21-s − 4·22-s − 4·23-s − 24-s + 25-s − 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.46958620523344, −13.02953813454079, −12.49827744846684, −12.16354574196700, −11.91314917011578, −11.08306056515639, −10.44901757201856, −10.31732947741011, −9.836263495832869, −9.458126228228338, −8.556397817841802, −8.227259283027857, −7.461374987659791, −6.940084178739232, −6.543549977873828, −6.147308367460801, −5.629429558543035, −4.993832336238571, −4.809235439620750, −3.963176435379017, −3.279082485275113, −2.994135031560669, −2.256341731540652, −1.759279613842497, −0.6401485172934032, 0,
0.6401485172934032, 1.759279613842497, 2.256341731540652, 2.994135031560669, 3.279082485275113, 3.963176435379017, 4.809235439620750, 4.993832336238571, 5.629429558543035, 6.147308367460801, 6.543549977873828, 6.940084178739232, 7.461374987659791, 8.227259283027857, 8.556397817841802, 9.458126228228338, 9.836263495832869, 10.31732947741011, 10.44901757201856, 11.08306056515639, 11.91314917011578, 12.16354574196700, 12.49827744846684, 13.02953813454079, 13.46958620523344