| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 8-s + 9-s + 4·10-s + 12-s + 13-s + 4·15-s + 16-s − 17-s + 18-s − 2·19-s + 4·20-s − 4·23-s + 24-s + 11·25-s + 26-s + 27-s − 2·29-s + 4·30-s + 32-s − 34-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s + 0.277·13-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.894·20-s − 0.834·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 0.730·30-s + 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.53296810025496, −13.01728025719091, −12.90031628351728, −12.33039711500195, −11.59234594748441, −11.19642444917080, −10.50577524016775, −10.17252941999898, −9.736016324688556, −9.333508649529292, −8.613576263720946, −8.386394955111233, −7.671077964875244, −6.922321960565114, −6.570250586593932, −6.187975439608015, −5.556291850624540, −5.156836031895313, −4.648089913449150, −3.891835445270439, −3.413839746778142, −2.713604897297680, −2.258227560422209, −1.669624547922325, −1.345772336058013, 0,
1.345772336058013, 1.669624547922325, 2.258227560422209, 2.713604897297680, 3.413839746778142, 3.891835445270439, 4.648089913449150, 5.156836031895313, 5.556291850624540, 6.187975439608015, 6.570250586593932, 6.922321960565114, 7.671077964875244, 8.386394955111233, 8.613576263720946, 9.333508649529292, 9.736016324688556, 10.17252941999898, 10.50577524016775, 11.19642444917080, 11.59234594748441, 12.33039711500195, 12.90031628351728, 13.01728025719091, 13.53296810025496
