L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 4·11-s + 13-s + 16-s + 6·17-s + 4·19-s − 20-s + 4·22-s − 8·23-s + 25-s − 26-s − 6·29-s − 8·31-s − 32-s − 6·34-s − 10·37-s − 4·38-s + 40-s + 6·41-s + 4·43-s − 4·44-s + 8·46-s − 7·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + 1.17·46-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.402437326353945088461831893248, −8.490403204098803911613209618504, −7.54788789198252757941968322284, −7.45586944398122020573570806182, −5.88475078175345459986396322172, −5.36101206913072414201885147070, −3.89256533853822824063870308537, −2.98523621593750721047361308357, −1.63917888763865576740316873481, 0,
1.63917888763865576740316873481, 2.98523621593750721047361308357, 3.89256533853822824063870308537, 5.36101206913072414201885147070, 5.88475078175345459986396322172, 7.45586944398122020573570806182, 7.54788789198252757941968322284, 8.490403204098803911613209618504, 9.402437326353945088461831893248