L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 2·13-s + 14-s + 16-s − 6·17-s + 4·19-s − 20-s + 25-s − 2·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 35-s − 2·37-s − 4·38-s + 40-s − 6·41-s − 8·43-s + 12·47-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s + 1/5·25-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 + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.74703122534574, −12.17770668799838, −11.89839219262965, −11.38976804133623, −10.99824328430131, −10.43707897616078, −10.22815486340732, −9.530605427325487, −9.116651828345916, −8.665443953684484, −8.407227062414642, −7.762596175082262, −7.282809881486823, −6.825412036577490, −6.478658958413733, −5.961403432535544, −5.243494686650567, −4.882296049059812, −4.170336713726470, −3.535485602201853, −3.339370808827270, −2.366496309720177, −2.180924748267799, −1.226733696313042, −0.7183973268201218, 0,
0.7183973268201218, 1.226733696313042, 2.180924748267799, 2.366496309720177, 3.339370808827270, 3.535485602201853, 4.170336713726470, 4.882296049059812, 5.243494686650567, 5.961403432535544, 6.478658958413733, 6.825412036577490, 7.282809881486823, 7.762596175082262, 8.407227062414642, 8.665443953684484, 9.116651828345916, 9.530605427325487, 10.22815486340732, 10.43707897616078, 10.99824328430131, 11.38976804133623, 11.89839219262965, 12.17770668799838, 12.74703122534574