L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s − 4·11-s + 2·13-s + 14-s − 16-s + 2·17-s − 4·19-s + 4·22-s − 2·26-s + 28-s − 29-s − 8·31-s − 5·32-s − 2·34-s + 10·37-s + 4·38-s + 6·41-s − 12·43-s + 4·44-s − 8·47-s + 49-s − 2·52-s + 6·53-s − 3·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s − 0.392·26-s + 0.188·28-s − 0.185·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.82·43-s + 0.603·44-s − 1.16·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4636654744\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4636654744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.69873332352908, −14.09043300974672, −13.39384978481490, −13.14722620589518, −12.72634779254843, −12.16282010434815, −11.17968662918982, −10.93754803503441, −10.42545105362292, −9.789154417598280, −9.480746745535211, −8.828298439266968, −8.305242773491981, −7.781002863305730, −7.493413237800720, −6.546518920003853, −6.110546149579394, −5.236710873112097, −4.974040866063541, −4.068935007049539, −3.610222015340557, −2.783951963282679, −2.024989216745849, −1.241343533690278, −0.2945948276081118,
0.2945948276081118, 1.241343533690278, 2.024989216745849, 2.783951963282679, 3.610222015340557, 4.068935007049539, 4.974040866063541, 5.236710873112097, 6.110546149579394, 6.546518920003853, 7.493413237800720, 7.781002863305730, 8.305242773491981, 8.828298439266968, 9.480746745535211, 9.789154417598280, 10.42545105362292, 10.93754803503441, 11.17968662918982, 12.16282010434815, 12.72634779254843, 13.14722620589518, 13.39384978481490, 14.09043300974672, 14.69873332352908