L(s) = 1 | + 3-s − 5-s + 9-s − 13-s − 15-s − 17-s − 4·19-s + 25-s + 27-s + 6·29-s + 8·31-s − 6·37-s − 39-s + 10·41-s − 4·43-s − 45-s − 4·47-s − 7·49-s − 51-s + 2·53-s − 4·57-s + 2·61-s + 65-s + 12·67-s + 14·73-s + 75-s − 8·79-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s − 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s − 49-s − 0.140·51-s + 0.274·53-s − 0.529·57-s + 0.256·61-s + 0.124·65-s + 1.46·67-s + 1.63·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09548561964561, −12.87178373824575, −12.30490312833773, −11.93407154874616, −11.35306108350644, −10.94076759907165, −10.31174399127694, −10.03941947926250, −9.453410780250979, −8.909499849421934, −8.498882491128115, −7.928923910063482, −7.842836988115013, −6.914716776121687, −6.578276808582860, −6.283086609021919, −5.262279219029397, −4.999033085883636, −4.239676065023777, −4.015670240024106, −3.271081200068474, −2.690257888742556, −2.306326634064683, −1.510226836106998, −0.8236225211633329, 0,
0.8236225211633329, 1.510226836106998, 2.306326634064683, 2.690257888742556, 3.271081200068474, 4.015670240024106, 4.239676065023777, 4.999033085883636, 5.262279219029397, 6.283086609021919, 6.578276808582860, 6.914716776121687, 7.842836988115013, 7.928923910063482, 8.498882491128115, 8.909499849421934, 9.453410780250979, 10.03941947926250, 10.31174399127694, 10.94076759907165, 11.35306108350644, 11.93407154874616, 12.30490312833773, 12.87178373824575, 13.09548561964561