| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 4·13-s − 14-s + 16-s − 17-s + 6·19-s − 5·25-s + 4·26-s − 28-s − 6·29-s − 4·31-s + 32-s − 34-s − 7·37-s + 6·38-s − 9·41-s − 12·47-s + 49-s − 5·50-s + 4·52-s + 11·53-s − 56-s − 6·58-s − 59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.37·19-s − 25-s + 0.784·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1.15·37-s + 0.973·38-s − 1.40·41-s − 1.75·47-s + 1/7·49-s − 0.707·50-s + 0.554·52-s + 1.51·53-s − 0.133·56-s − 0.787·58-s − 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.14749881976795, −12.74246256980074, −12.00392435685880, −11.69757556341226, −11.44442750795782, −10.74009708989159, −10.43731275056302, −9.774901773358889, −9.443766574136794, −8.862375192329935, −8.333090424965587, −7.862267281804773, −7.286334143730513, −6.805067497375874, −6.480684191452120, −5.636206048074887, −5.545223957293615, −5.029803939666793, −4.200287388451514, −3.765233615139658, −3.358674501395402, −2.956463560281699, −1.906855768493576, −1.762227640632602, −0.8583390744206093, 0,
0.8583390744206093, 1.762227640632602, 1.906855768493576, 2.956463560281699, 3.358674501395402, 3.765233615139658, 4.200287388451514, 5.029803939666793, 5.545223957293615, 5.636206048074887, 6.480684191452120, 6.805067497375874, 7.286334143730513, 7.862267281804773, 8.333090424965587, 8.862375192329935, 9.443766574136794, 9.774901773358889, 10.43731275056302, 10.74009708989159, 11.44442750795782, 11.69757556341226, 12.00392435685880, 12.74246256980074, 13.14749881976795
