| L(s) = 1 | − 2·2-s + 2·4-s − 13-s − 4·16-s + 3·17-s − 4·19-s − 8·23-s + 2·26-s − 6·29-s − 2·31-s + 8·32-s − 6·34-s − 4·37-s + 8·38-s − 8·41-s − 4·43-s + 16·46-s − 6·47-s − 7·49-s − 2·52-s − 5·53-s + 12·58-s − 10·59-s + 2·61-s + 4·62-s − 8·64-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.277·13-s − 16-s + 0.727·17-s − 0.917·19-s − 1.66·23-s + 0.392·26-s − 1.11·29-s − 0.359·31-s + 1.41·32-s − 1.02·34-s − 0.657·37-s + 1.29·38-s − 1.24·41-s − 0.609·43-s + 2.35·46-s − 0.875·47-s − 49-s − 0.277·52-s − 0.686·53-s + 1.57·58-s − 1.30·59-s + 0.256·61-s + 0.508·62-s − 64-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 353925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 353925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.77959066526818, −12.12265909150590, −11.68236190466156, −11.41544036476011, −10.69328317056627, −10.36100782696239, −10.08712048008872, −9.457119051522378, −9.301395824737716, −8.539252878786364, −8.296147011421537, −7.830897853360221, −7.439652526728764, −6.923114259342303, −6.367551786186085, −5.982021802665359, −5.324261238793399, −4.718638625386961, −4.312874202588943, −3.510782642894028, −3.210258983414118, −2.250135512949966, −1.741158783138149, −1.576612974805027, −0.4677820497179424, 0,
0.4677820497179424, 1.576612974805027, 1.741158783138149, 2.250135512949966, 3.210258983414118, 3.510782642894028, 4.312874202588943, 4.718638625386961, 5.324261238793399, 5.982021802665359, 6.367551786186085, 6.923114259342303, 7.439652526728764, 7.830897853360221, 8.296147011421537, 8.539252878786364, 9.301395824737716, 9.457119051522378, 10.08712048008872, 10.36100782696239, 10.69328317056627, 11.41544036476011, 11.68236190466156, 12.12265909150590, 12.77959066526818
