L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s − 2·9-s + 2·11-s − 12-s − 13-s + 16-s + 3·17-s + 2·18-s + 19-s − 2·22-s + 23-s + 24-s + 26-s + 5·27-s − 5·29-s + 8·31-s − 32-s − 2·33-s − 3·34-s − 2·36-s + 2·37-s − 38-s + 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.196·26-s + 0.962·27-s − 0.928·29-s + 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.514·34-s − 1/3·36-s + 0.328·37-s − 0.162·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.74108276493588, −14.53421147720038, −13.87240196498571, −13.37583294567657, −12.48791361010969, −12.19392411673645, −11.71838104499947, −11.19656188975435, −10.76202272157541, −10.19639867567057, −9.576003526875155, −9.169780519631086, −8.627578167771810, −7.898257541112964, −7.600424419464534, −6.804445730122626, −6.325206558602091, −5.768988196509251, −5.290589284108260, −4.517585457147676, −3.832443689511285, −2.971367976359002, −2.553581712219344, −1.494141952428425, −0.8927802018116104, 0,
0.8927802018116104, 1.494141952428425, 2.553581712219344, 2.971367976359002, 3.832443689511285, 4.517585457147676, 5.290589284108260, 5.768988196509251, 6.325206558602091, 6.804445730122626, 7.600424419464534, 7.898257541112964, 8.627578167771810, 9.169780519631086, 9.576003526875155, 10.19639867567057, 10.76202272157541, 11.19656188975435, 11.71838104499947, 12.19392411673645, 12.48791361010969, 13.37583294567657, 13.87240196498571, 14.53421147720038, 14.74108276493588