L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s + 6·17-s + 4·19-s + 2·26-s + 28-s − 6·29-s − 4·31-s + 32-s + 6·34-s − 2·37-s + 4·38-s − 6·41-s − 4·43-s + 49-s + 2·52-s − 6·53-s + 56-s − 6·58-s − 12·59-s − 14·61-s − 4·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.328·37-s + 0.648·38-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.277·52-s − 0.824·53-s + 0.133·56-s − 0.787·58-s − 1.56·59-s − 1.79·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.56162285832409, −12.28366434736204, −11.93370859337872, −11.33760636750627, −11.01435121301692, −10.60672615242156, −10.01620396138792, −9.541295005650906, −9.203040307364117, −8.494173157388071, −8.031450534440076, −7.569416264517841, −7.314841417707664, −6.590940221905779, −6.154409072141296, −5.641761518568120, −5.144338727445521, −4.962401710208487, −4.138042161288949, −3.601942645582952, −3.311109591165443, −2.804853440825668, −1.877241669629544, −1.572062410156196, −0.9509640496116435, 0,
0.9509640496116435, 1.572062410156196, 1.877241669629544, 2.804853440825668, 3.311109591165443, 3.601942645582952, 4.138042161288949, 4.962401710208487, 5.144338727445521, 5.641761518568120, 6.154409072141296, 6.590940221905779, 7.314841417707664, 7.569416264517841, 8.031450534440076, 8.494173157388071, 9.203040307364117, 9.541295005650906, 10.01620396138792, 10.60672615242156, 11.01435121301692, 11.33760636750627, 11.93370859337872, 12.28366434736204, 12.56162285832409