L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s − 12-s − 2·13-s + 4·14-s + 16-s − 6·17-s + 18-s − 4·19-s − 4·21-s − 24-s − 2·26-s − 27-s + 4·28-s − 6·29-s + 8·31-s + 32-s − 6·34-s + 36-s − 2·37-s − 4·38-s + 2·39-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 − 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.872·21-s − 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.499036832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499036832\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.01009824051402823202590914616, −11.91033651235536617108337917161, −11.23148855774643011734788423771, −10.40288135541038540368452071068, −8.749862280172923617009160642326, −7.56689807773085737112376729195, −6.38949936725272530282824625186, −5.07079211702149946545201877286, −4.30912662284802770612117339996, −2.07452981833800877942993394416,
2.07452981833800877942993394416, 4.30912662284802770612117339996, 5.07079211702149946545201877286, 6.38949936725272530282824625186, 7.56689807773085737112376729195, 8.749862280172923617009160642326, 10.40288135541038540368452071068, 11.23148855774643011734788423771, 11.91033651235536617108337917161, 13.01009824051402823202590914616