| L(s) = 1 | − 2·2-s + 4·4-s − 32·7-s − 8·8-s + 60·11-s + 34·13-s + 64·14-s + 16·16-s + 42·17-s − 76·19-s − 120·22-s − 68·26-s − 128·28-s − 6·29-s − 232·31-s − 32·32-s − 84·34-s − 134·37-s + 152·38-s − 234·41-s + 412·43-s + 240·44-s − 360·47-s + 681·49-s + 136·52-s + 222·53-s + 256·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.72·7-s − 0.353·8-s + 1.64·11-s + 0.725·13-s + 1.22·14-s + 1/4·16-s + 0.599·17-s − 0.917·19-s − 1.16·22-s − 0.512·26-s − 0.863·28-s − 0.0384·29-s − 1.34·31-s − 0.176·32-s − 0.423·34-s − 0.595·37-s + 0.648·38-s − 0.891·41-s + 1.46·43-s + 0.822·44-s − 1.11·47-s + 1.98·49-s + 0.362·52-s + 0.575·53-s + 0.610·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 + 234 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 + 490 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 746 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 + 804 T + p^{3} T^{2} \) |
| 89 | \( 1 - 678 T + p^{3} T^{2} \) |
| 97 | \( 1 + 2 p T + p^{3} T^{2} \) |
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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
−10.08091160105148083731026840322, −9.214680108121570066410073116053, −8.790803481919396086710574280756, −7.35465707811293591364154997623, −6.49031981164757099297090506928, −5.94562703525766440971659985278, −3.98311266602636069799208741447, −3.15803445145612579308496980617, −1.48177681066133768791967466348, 0,
1.48177681066133768791967466348, 3.15803445145612579308496980617, 3.98311266602636069799208741447, 5.94562703525766440971659985278, 6.49031981164757099297090506928, 7.35465707811293591364154997623, 8.790803481919396086710574280756, 9.214680108121570066410073116053, 10.08091160105148083731026840322
