L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 6·11-s + 12-s + 13-s + 14-s + 16-s + 18-s − 4·19-s + 21-s − 6·22-s − 3·23-s + 24-s + 26-s + 27-s + 28-s − 3·29-s − 5·31-s + 32-s − 6·33-s + 36-s − 10·37-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 − 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.898·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.609229260\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609229260\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.74545694840307, −12.36590701192696, −11.92804430436805, −11.19435402051744, −10.85844031987531, −10.50407212904580, −10.05777785047196, −9.526191961465698, −8.867716270601296, −8.319331174338111, −8.132707329011851, −7.577432280796870, −7.079627688835201, −6.650972087991846, −5.945249604294631, −5.466703949580421, −5.095138688959806, −4.589360025037038, −4.006098191863684, −3.430708081497057, −3.069774131101353, −2.291584859058262, −2.000369582144807, −1.441757788612557, −0.2549111807672670,
0.2549111807672670, 1.441757788612557, 2.000369582144807, 2.291584859058262, 3.069774131101353, 3.430708081497057, 4.006098191863684, 4.589360025037038, 5.095138688959806, 5.466703949580421, 5.945249604294631, 6.650972087991846, 7.079627688835201, 7.577432280796870, 8.132707329011851, 8.319331174338111, 8.867716270601296, 9.526191961465698, 10.05777785047196, 10.50407212904580, 10.85844031987531, 11.19435402051744, 11.92804430436805, 12.36590701192696, 12.74545694840307