L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 3·13-s − 14-s + 15-s + 16-s − 4·17-s − 18-s + 6·19-s − 20-s − 21-s + 22-s + 24-s + 25-s + 3·26-s − 27-s + 28-s − 30-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 − 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.188·28-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6422468167\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6422468167\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.18552969418627, −12.44433761116267, −12.07354042824699, −11.77845430173203, −11.33612545217397, −10.78671320227129, −10.30174759572741, −10.03706397202572, −9.359922812104912, −8.894531504559291, −8.393361286803176, −7.930702694115566, −7.239272391830431, −7.088832146438384, −6.615855025891970, −5.713570349934911, −5.430813880273601, −4.826101502327948, −4.339941087928643, −3.657529434780876, −2.952568498226631, −2.418174454983383, −1.712338061796264, −1.045547577543593, −0.3045753784737951,
0.3045753784737951, 1.045547577543593, 1.712338061796264, 2.418174454983383, 2.952568498226631, 3.657529434780876, 4.339941087928643, 4.826101502327948, 5.430813880273601, 5.713570349934911, 6.615855025891970, 7.088832146438384, 7.239272391830431, 7.930702694115566, 8.393361286803176, 8.894531504559291, 9.359922812104912, 10.03706397202572, 10.30174759572741, 10.78671320227129, 11.33612545217397, 11.77845430173203, 12.07354042824699, 12.44433761116267, 13.18552969418627