L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 6·11-s + 2·13-s + 14-s + 16-s + 2·17-s + 4·19-s − 6·22-s + 4·23-s − 2·26-s − 28-s + 2·29-s − 2·31-s − 32-s − 2·34-s − 10·37-s − 4·38-s + 6·41-s − 2·43-s + 6·44-s − 4·46-s − 2·47-s + 49-s + 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 1.27·22-s + 0.834·23-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.359·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s − 0.648·38-s + 0.937·41-s − 0.304·43-s + 0.904·44-s − 0.589·46-s − 0.291·47-s + 1/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.558097563\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.558097563\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.950163787754823663326790518302, −7.993950275021705755190178558370, −7.15916677784730730825443278472, −6.56855266218116599245359312977, −5.89871980785880398996070256741, −4.86546965928682691873500886567, −3.69583531568847373477475719618, −3.16992773279632385138876218464, −1.72545102894322634760289250934, −0.903111156649165609200585998870,
0.903111156649165609200585998870, 1.72545102894322634760289250934, 3.16992773279632385138876218464, 3.69583531568847373477475719618, 4.86546965928682691873500886567, 5.89871980785880398996070256741, 6.56855266218116599245359312977, 7.15916677784730730825443278472, 7.993950275021705755190178558370, 8.950163787754823663326790518302