| L(s) = 1 | + 2·2-s − 4·4-s − 7·7-s − 24·8-s + 21·11-s + 24·13-s − 14·14-s − 16·16-s + 22·17-s + 16·19-s + 42·22-s + 25·23-s + 48·26-s + 28·28-s − 167·29-s + 10·31-s + 160·32-s + 44·34-s − 133·37-s + 32·38-s + 168·41-s − 97·43-s − 84·44-s + 50·46-s + 400·47-s + 49·49-s − 96·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s + 0.575·11-s + 0.512·13-s − 0.267·14-s − 1/4·16-s + 0.313·17-s + 0.193·19-s + 0.407·22-s + 0.226·23-s + 0.362·26-s + 0.188·28-s − 1.06·29-s + 0.0579·31-s + 0.883·32-s + 0.221·34-s − 0.590·37-s + 0.136·38-s + 0.639·41-s − 0.344·43-s − 0.287·44-s + 0.160·46-s + 1.24·47-s + 1/7·49-s − 0.256·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - 21 T + p^{3} T^{2} \) |
| 13 | \( 1 - 24 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 - 25 T + p^{3} T^{2} \) |
| 29 | \( 1 + 167 T + p^{3} T^{2} \) |
| 31 | \( 1 - 10 T + p^{3} T^{2} \) |
| 37 | \( 1 + 133 T + p^{3} T^{2} \) |
| 41 | \( 1 - 168 T + p^{3} T^{2} \) |
| 43 | \( 1 + 97 T + p^{3} T^{2} \) |
| 47 | \( 1 - 400 T + p^{3} T^{2} \) |
| 53 | \( 1 - 182 T + p^{3} T^{2} \) |
| 59 | \( 1 + 488 T + p^{3} T^{2} \) |
| 61 | \( 1 - 28 T + p^{3} T^{2} \) |
| 67 | \( 1 + 967 T + p^{3} T^{2} \) |
| 71 | \( 1 - 285 T + p^{3} T^{2} \) |
| 73 | \( 1 + 838 T + p^{3} T^{2} \) |
| 79 | \( 1 + 469 T + p^{3} T^{2} \) |
| 83 | \( 1 - 406 T + p^{3} T^{2} \) |
| 89 | \( 1 + 324 T + p^{3} T^{2} \) |
| 97 | \( 1 + 114 T + p^{3} T^{2} \) |
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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
−8.904320674399158517361662762875, −7.85369165606856364087296905227, −6.89108593358422050566743397642, −5.98940927126520713769637198745, −5.40965870169922345916075684343, −4.32816678425902887268717847919, −3.68571478910013389536919597536, −2.80593585171958317572250452527, −1.29815158156138756505394464092, 0,
1.29815158156138756505394464092, 2.80593585171958317572250452527, 3.68571478910013389536919597536, 4.32816678425902887268717847919, 5.40965870169922345916075684343, 5.98940927126520713769637198745, 6.89108593358422050566743397642, 7.85369165606856364087296905227, 8.904320674399158517361662762875
