| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s + 11-s − 2·14-s − 16-s − 2·19-s − 22-s − 23-s − 5·25-s − 2·28-s + 10·29-s − 4·31-s − 5·32-s − 2·37-s + 2·38-s − 2·41-s + 2·43-s − 44-s + 46-s − 8·47-s − 3·49-s + 5·50-s + 4·53-s + 6·56-s − 10·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 0.301·11-s − 0.534·14-s − 1/4·16-s − 0.458·19-s − 0.213·22-s − 0.208·23-s − 25-s − 0.377·28-s + 1.85·29-s − 0.718·31-s − 0.883·32-s − 0.328·37-s + 0.324·38-s − 0.312·41-s + 0.304·43-s − 0.150·44-s + 0.147·46-s − 1.16·47-s − 3/7·49-s + 0.707·50-s + 0.549·53-s + 0.801·56-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384813 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384813 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−12.58323607433383, −12.28411410390097, −11.67060367384877, −11.28844159934391, −10.81573071783408, −10.34011813872994, −9.960537656267826, −9.559680382723374, −8.958701346184723, −8.614869096365025, −8.252635812845848, −7.751038995863871, −7.417007406586974, −6.814470540861097, −6.182715195585172, −5.853617830962476, −5.039038428569650, −4.653998135290027, −4.457485347332922, −3.605262794362739, −3.312807778732797, −2.345963751681424, −1.840051238418137, −1.377854977284661, −0.6895473417701013, 0,
0.6895473417701013, 1.377854977284661, 1.840051238418137, 2.345963751681424, 3.312807778732797, 3.605262794362739, 4.457485347332922, 4.653998135290027, 5.039038428569650, 5.853617830962476, 6.182715195585172, 6.814470540861097, 7.417007406586974, 7.751038995863871, 8.252635812845848, 8.614869096365025, 8.958701346184723, 9.559680382723374, 9.960537656267826, 10.34011813872994, 10.81573071783408, 11.28844159934391, 11.67060367384877, 12.28411410390097, 12.58323607433383
