L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 11-s + 12-s − 13-s − 2·14-s + 16-s − 17-s − 18-s + 3·19-s + 2·21-s − 22-s + 23-s − 24-s + 26-s + 27-s + 2·28-s + 29-s + 2·31-s − 32-s + 33-s + 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.688·19-s + 0.436·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.185·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.75672392142892, −12.06368559911937, −11.84128350740106, −11.35944992067328, −10.89195544559441, −10.38910971339112, −9.988610693206407, −9.421447000954016, −9.215285653498812, −8.492004328720647, −8.255038679005752, −7.896878212278400, −7.191447831038433, −6.980600995993992, −6.438936687277169, −5.755397318041961, −5.284388292593810, −4.620818881517441, −4.373334417933336, −3.425519931393616, −3.191187024210005, −2.522413427679531, −1.832226949044690, −1.529714976617373, −0.8395771636647806, 0,
0.8395771636647806, 1.529714976617373, 1.832226949044690, 2.522413427679531, 3.191187024210005, 3.425519931393616, 4.373334417933336, 4.620818881517441, 5.284388292593810, 5.755397318041961, 6.438936687277169, 6.980600995993992, 7.191447831038433, 7.896878212278400, 8.255038679005752, 8.492004328720647, 9.215285653498812, 9.421447000954016, 9.988610693206407, 10.38910971339112, 10.89195544559441, 11.35944992067328, 11.84128350740106, 12.06368559911937, 12.75672392142892