L(s) = 1 | + 3-s − 5-s − 5·7-s + 9-s − 11-s + 13-s − 15-s + 3·17-s + 6·19-s − 5·21-s − 3·23-s + 25-s + 27-s + 4·29-s − 33-s + 5·35-s − 5·37-s + 39-s + 11·41-s − 6·43-s − 45-s + 18·49-s + 3·51-s − 9·53-s + 55-s + 6·57-s − 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 1.37·19-s − 1.09·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.174·33-s + 0.845·35-s − 0.821·37-s + 0.160·39-s + 1.71·41-s − 0.914·43-s − 0.149·45-s + 18/7·49-s + 0.420·51-s − 1.23·53-s + 0.134·55-s + 0.794·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.62351450688618062134783975535, −7.12723176355176202393670904365, −6.28311537006880299798680585076, −5.73690722485889775373333119997, −4.69724021505541570659022099184, −3.72044664303005925848728513234, −3.22202097433461828467957235376, −2.67741321325165252650277444273, −1.23158835594670790442412835349, 0,
1.23158835594670790442412835349, 2.67741321325165252650277444273, 3.22202097433461828467957235376, 3.72044664303005925848728513234, 4.69724021505541570659022099184, 5.73690722485889775373333119997, 6.28311537006880299798680585076, 7.12723176355176202393670904365, 7.62351450688618062134783975535