L(s) = 1 | − 2-s − 2·3-s − 4-s − 2·5-s + 2·6-s − 5·7-s + 3·8-s + 9-s + 2·10-s − 11-s + 2·12-s − 4·13-s + 5·14-s + 4·15-s − 16-s − 3·17-s − 18-s − 5·19-s + 2·20-s + 10·21-s + 22-s − 3·23-s − 6·24-s − 25-s + 4·26-s + 4·27-s + 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.88·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s + 1.33·14-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.14·19-s + 0.447·20-s + 2.18·21-s + 0.213·22-s − 0.625·23-s − 1.22·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26743 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26743 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 47 | \( 1 + T \) |
| 569 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.19749625463088, −15.85654962258102, −15.14143471881911, −14.70392145858624, −13.71990123031011, −13.29271839244716, −12.62719526714702, −12.43538739632992, −11.82570406199275, −11.04352352173562, −10.69504464094628, −10.06166800634527, −9.676412020054801, −9.032835061361824, −8.542735821015412, −7.757742741647074, −7.149128089039270, −6.763154246286237, −6.034191041530998, −5.448585386097418, −4.760645213309887, −4.062023725474041, −3.581846771758994, −2.655036109386631, −1.660589697446375, 0, 0, 0,
1.660589697446375, 2.655036109386631, 3.581846771758994, 4.062023725474041, 4.760645213309887, 5.448585386097418, 6.034191041530998, 6.763154246286237, 7.149128089039270, 7.757742741647074, 8.542735821015412, 9.032835061361824, 9.676412020054801, 10.06166800634527, 10.69504464094628, 11.04352352173562, 11.82570406199275, 12.43538739632992, 12.62719526714702, 13.29271839244716, 13.71990123031011, 14.70392145858624, 15.14143471881911, 15.85654962258102, 16.19749625463088