L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 4·13-s − 14-s + 16-s + 4·19-s + 4·26-s + 28-s − 6·29-s − 10·31-s − 32-s − 2·37-s − 4·38-s − 12·41-s − 4·43-s + 6·47-s + 49-s − 4·52-s − 6·53-s − 56-s + 6·58-s + 6·59-s + 4·61-s + 10·62-s + 64-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 + 0.917·19-s + 0.784·26-s + 0.188·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s − 0.328·37-s − 0.648·38-s − 1.87·41-s − 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.554·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 0.781·59-s + 0.512·61-s + 1.27·62-s + 1/8·64-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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.89287597150328, −12.30088641467759, −11.97709611829003, −11.50491974402947, −11.11979551672191, −10.67983448942469, −10.06462701363990, −9.767640606524070, −9.407405545586981, −8.741713716001662, −8.538739096122949, −7.833517131211414, −7.455844266476187, −7.036421551904327, −6.802010619005555, −5.894186945857543, −5.348185134518645, −5.299955815796129, −4.469330086284424, −3.908486707276805, −3.304377858609090, −2.839471710060365, −2.125892003544475, −1.676047419082754, −1.172861417169623, 0, 0,
1.172861417169623, 1.676047419082754, 2.125892003544475, 2.839471710060365, 3.304377858609090, 3.908486707276805, 4.469330086284424, 5.299955815796129, 5.348185134518645, 5.894186945857543, 6.802010619005555, 7.036421551904327, 7.455844266476187, 7.833517131211414, 8.538739096122949, 8.741713716001662, 9.407405545586981, 9.767640606524070, 10.06462701363990, 10.67983448942469, 11.11979551672191, 11.50491974402947, 11.97709611829003, 12.30088641467759, 12.89287597150328