| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s − 4·13-s + 16-s − 6·17-s − 18-s − 8·19-s + 2·22-s + 23-s + 24-s + 4·26-s − 27-s + 4·29-s − 32-s + 2·33-s + 6·34-s + 36-s + 2·37-s + 8·38-s + 4·39-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.83·19-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.176·32-s + 0.348·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 1.29·38-s + 0.640·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5394908177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5394908177\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.591556920910459989337540257739, −7.951083160282523591332931975508, −6.94342275656798629514371947037, −6.64557875219879895552398018651, −5.66654127407213124932797278900, −4.79124720691970822537754330640, −4.10569325074294583770187706757, −2.62297255271741741004507829590, −2.04508758036819954688449020522, −0.46933715668305309924262465445,
0.46933715668305309924262465445, 2.04508758036819954688449020522, 2.62297255271741741004507829590, 4.10569325074294583770187706757, 4.79124720691970822537754330640, 5.66654127407213124932797278900, 6.64557875219879895552398018651, 6.94342275656798629514371947037, 7.951083160282523591332931975508, 8.591556920910459989337540257739
