L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 2·11-s + 13-s − 2·14-s + 16-s + 2·17-s − 4·19-s − 2·22-s + 26-s − 2·28-s − 4·29-s + 8·31-s + 32-s + 2·34-s − 6·37-s − 4·38-s + 6·41-s − 4·43-s − 2·44-s − 8·47-s − 3·49-s + 52-s + 2·53-s − 2·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.603·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.426·22-s + 0.196·26-s − 0.377·28-s − 0.742·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.986·37-s − 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.301·44-s − 1.16·47-s − 3/7·49-s + 0.138·52-s + 0.274·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.75895769889950141770720055671, −6.77153372548768128182547904443, −6.34685478800674956059786856134, −5.58782284507518106759486426670, −4.85127735949262231391218105283, −4.04312501686402339841895609473, −3.25126981652130652906895848842, −2.59306196361380656591048270734, −1.50298144790677844555176641546, 0,
1.50298144790677844555176641546, 2.59306196361380656591048270734, 3.25126981652130652906895848842, 4.04312501686402339841895609473, 4.85127735949262231391218105283, 5.58782284507518106759486426670, 6.34685478800674956059786856134, 6.77153372548768128182547904443, 7.75895769889950141770720055671