L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 11-s − 12-s − 2·13-s − 14-s + 16-s − 6·17-s + 18-s − 4·19-s + 21-s − 22-s − 24-s − 2·26-s − 27-s − 28-s + 6·29-s − 4·31-s + 32-s + 33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.213·22-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.733244102\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.733244102\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.24094856539612, −15.82263479342289, −15.40384903063654, −14.69630074458228, −14.18172722981278, −13.43757308488470, −12.96606123562719, −12.48620271340465, −12.00715596392984, −11.11933204937480, −10.89884647298205, −10.20887083676233, −9.520405992127324, −8.804781473386928, −8.097996211710850, −7.222628639547431, −6.774188377982718, −6.145673742717409, −5.571926264095061, −4.621772451856338, −4.428892710199122, −3.435686263944906, −2.537850013319081, −1.916975017888047, −0.5311669480212529,
0.5311669480212529, 1.916975017888047, 2.537850013319081, 3.435686263944906, 4.428892710199122, 4.621772451856338, 5.571926264095061, 6.145673742717409, 6.774188377982718, 7.222628639547431, 8.097996211710850, 8.804781473386928, 9.520405992127324, 10.20887083676233, 10.89884647298205, 11.11933204937480, 12.00715596392984, 12.48620271340465, 12.96606123562719, 13.43757308488470, 14.18172722981278, 14.69630074458228, 15.40384903063654, 15.82263479342289, 16.24094856539612