L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 5·11-s − 12-s + 4·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s − 6·19-s − 20-s + 21-s + 5·22-s + 6·23-s + 24-s + 25-s − 4·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.03933449182112, −14.72966608427871, −13.68841370778568, −13.19712321220026, −12.83721907866773, −12.47357222512488, −11.49788967801717, −11.41023489059745, −10.64319387602880, −10.43765313883405, −9.973760402419754, −9.055909265728482, −8.749035089367970, −8.138851084200000, −7.683994536918983, −6.910449038625806, −6.743994068156349, −5.811836935357164, −5.474008173418694, −4.844847097348249, −3.999117362731977, −3.381409782079104, −2.820908925945954, −1.888438429175949, −1.266765722791829, 0, 0,
1.266765722791829, 1.888438429175949, 2.820908925945954, 3.381409782079104, 3.999117362731977, 4.844847097348249, 5.474008173418694, 5.811836935357164, 6.743994068156349, 6.910449038625806, 7.683994536918983, 8.138851084200000, 8.749035089367970, 9.055909265728482, 9.973760402419754, 10.43765313883405, 10.64319387602880, 11.41023489059745, 11.49788967801717, 12.47357222512488, 12.83721907866773, 13.19712321220026, 13.68841370778568, 14.72966608427871, 15.03933449182112