| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 2·14-s + 16-s − 6·17-s − 4·19-s − 2·28-s − 2·29-s + 8·31-s − 32-s + 6·34-s − 8·37-s + 4·38-s − 6·41-s + 2·43-s − 8·47-s − 3·49-s − 6·53-s + 2·56-s + 2·58-s − 6·59-s − 6·61-s − 8·62-s + 64-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.377·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s + 0.648·38-s − 0.937·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.267·56-s + 0.262·58-s − 0.781·59-s − 0.768·61-s − 1.01·62-s + 1/8·64-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.63920428736361, −13.99171829986203, −13.53508745239707, −13.02558724440952, −12.52230353772739, −12.12657892296564, −11.29935052457433, −11.13338888845499, −10.46180227823231, −9.966530902628110, −9.586888193597321, −8.931266998075954, −8.520688309756321, −8.131747848557522, −7.355226817658349, −6.737180482768479, −6.495483644838349, −6.001325656026222, −5.129569543096925, −4.566153175947547, −3.933646642413489, −3.185250325816459, −2.668435760384834, −1.939606325597580, −1.329509506753985, 0, 0,
1.329509506753985, 1.939606325597580, 2.668435760384834, 3.185250325816459, 3.933646642413489, 4.566153175947547, 5.129569543096925, 6.001325656026222, 6.495483644838349, 6.737180482768479, 7.355226817658349, 8.131747848557522, 8.520688309756321, 8.931266998075954, 9.586888193597321, 9.966530902628110, 10.46180227823231, 11.13338888845499, 11.29935052457433, 12.12657892296564, 12.52230353772739, 13.02558724440952, 13.53508745239707, 13.99171829986203, 14.63920428736361
