L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 2·7-s − 8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·14-s + 2·15-s + 16-s + 6·17-s − 18-s + 4·19-s − 2·20-s + 2·21-s + 2·23-s + 24-s − 25-s + 2·26-s − 27-s − 2·28-s + 2·29-s − 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.436·21-s + 0.417·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7521474142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7521474142\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 1667 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.64942739825903, −16.11092736271900, −15.68091902001925, −15.25251210081259, −14.37038809994195, −13.88448439147342, −12.88449050223939, −12.41647761333513, −11.86182420097089, −11.53851239215092, −10.70344124341372, −10.12376897912923, −9.631131961542838, −9.088329612193720, −8.029179529252612, −7.736476447026688, −7.123346671599011, −6.389634549239950, −5.755685269735279, −4.976712093769295, −4.141042598995585, −3.268184081999317, −2.725053667828908, −1.305573526178041, −0.5237333088786354,
0.5237333088786354, 1.305573526178041, 2.725053667828908, 3.268184081999317, 4.141042598995585, 4.976712093769295, 5.755685269735279, 6.389634549239950, 7.123346671599011, 7.736476447026688, 8.029179529252612, 9.088329612193720, 9.631131961542838, 10.12376897912923, 10.70344124341372, 11.53851239215092, 11.86182420097089, 12.41647761333513, 12.88449050223939, 13.88448439147342, 14.37038809994195, 15.25251210081259, 15.68091902001925, 16.11092736271900, 16.64942739825903