L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s + 11-s + 12-s − 4·13-s + 3·14-s + 16-s + 17-s − 18-s + 4·19-s − 3·21-s − 22-s − 7·23-s − 24-s + 4·26-s + 27-s − 3·28-s − 7·29-s + 7·31-s − 32-s + 33-s − 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.801·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.654·21-s − 0.213·22-s − 1.45·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.566·28-s − 1.29·29-s + 1.25·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−15.61322859694506, −14.97753153335476, −14.50819439524942, −13.87203117874185, −13.48688005797986, −12.65609380761981, −12.30748682882495, −11.81603069413719, −11.16735865752333, −10.23183248713152, −10.04075517049170, −9.464982208670348, −9.162086860912282, −8.382537404172050, −7.761637068701267, −7.313712404920985, −6.793442232206959, −6.064708300503480, −5.551609777006508, −4.633223382154799, −3.797400796148851, −3.311467074361504, −2.531729284386285, −2.003104463033481, −0.9261951068501417, 0,
0.9261951068501417, 2.003104463033481, 2.531729284386285, 3.311467074361504, 3.797400796148851, 4.633223382154799, 5.551609777006508, 6.064708300503480, 6.793442232206959, 7.313712404920985, 7.761637068701267, 8.382537404172050, 9.162086860912282, 9.464982208670348, 10.04075517049170, 10.23183248713152, 11.16735865752333, 11.81603069413719, 12.30748682882495, 12.65609380761981, 13.48688005797986, 13.87203117874185, 14.50819439524942, 14.97753153335476, 15.61322859694506